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Smith and Critical Groups of Graphs

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Title:
Smith and Critical Groups of Graphs
Creator:
Pantangi, Venkata Raghutej
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (90 p.)

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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mathematics
Committee Chair:
Sin,Peter K
Committee Co-Chair:
Turull,Alexandre
Committee Members:
Crew,Richard Malcolm
Keating,Kevin P
Sitharam,Meera
Graduation Date:
5/3/2019

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Subjects / Keywords:
laplacian
Mathematics -- Dissertations, Academic -- UF
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
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Mathematics thesis, Ph.D.

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Abstract:
In this Dissertation, we compute the elementary divisors of the adjacency and Laplacian matrices of two families of strongly regular graphs: Polar graphs and van Lint-Schrijver Cyclotomic Strongly Regular Graphs. ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2019.
Local:
Adviser: Sin,Peter K.
Local:
Co-adviser: Turull,Alexandre.
Statement of Responsibility:
by Venkata Raghutej Pantangi.

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UFRGP
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Applicable rights reserved.
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LD1780 2019 ( lcc )

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SMITHANDCRITICALGROUPSOFGRAPHS By VENKATARAGHUTEJPANTANGI ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2019

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c 2019VenkataRaghuTejPantangi

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Idedicatethistomyfamily,friends,andteachers.

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ACKNOWLEDGMENTS ImustbeginbythankingmyadvisorPeterSin.Thisworkwouldnothavebeenpossible withouthispatience,guidance,andsupportforthelastveyears.Iwouldalsoliketothank therestofmysupervisorycommitteefortheircommentsandkindwords:Dr.Alexandre Turull,Dr.KevinKeating,Dr.RichardCrew,Dr.MeeraSitharaman,andDr.AndrewVince. Iwishtothankallmyteachersatundergraduatelevel.IamespeciallyindebtedtoDr. AnupamKumarSinghandDr.AmritanshuPrasad.Theyintroducedmetothewonderful topicsofRepresentationtheoryandrelatedcombinatorics. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS...................................4 LISTOFTABLES.......................................7 LISTOFFIGURES......................................8 ABSTRACT..........................................9 CHAPTER 1INTRODUCTION....................................10 2PRELIMINARIESANDDEFINITIONS.........................12 2.1SmithNormalForm................................12 2.2IncidenceMatrices,SmithNormalForms,andRepresentationTheory.....15 2.3GraphsandMatrices................................15 2.4SmithandCriticalGroupsofGraphs.......................16 2.4.1ChipFiringGame.............................17 2.4.2FamiliesofGraphsWithKnown Critical Groups.............19 2.5StronglyRegularGraphs.............................19 2.6Rank 3 PermutationGroupsandStronglyRegularGraphs............21 3RESULTS........................................22 3.1PolarGraphs...................................22 3.2VanLint-SchrijverGraphs.............................29 4PROOFSOFTHEOREM13ANDTHEOREM14...................35 4.1Nilpotenceof A and K Modulo ` ..........................35 4.2When A ` and L ` AreNotNilpotent........................37 4.2.1ElementaryDivisorsof S .........................37 4.2.2ElementaryDivisorsof K .........................38 4.3When A ` and L ` AreNilpotent...........................40 4.4 2 -ElementaryDivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.97 -1.793 Td [(s q ; m .............42 4.4.1SubmoduleStructure...........................43 4.4.2 2 -ElementaryDivisorswhen m IsEven..................43 4.4.3 2 -ElementaryDivisorswhen m IsOdd..................44 4.5 2 -ElementaryDivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.571 -1.794 Td [(o q ; m .............45 4.5.1SubmoduleStructure...........................45 4.5.2 2 -ElementaryDivisorswhen m IsEven..................46 4.5.3 2 -ElementaryDivisorsof S and K ,when m IsOdd............48 4.6 ` -ElementaryDivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.571 -1.793 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.793 Td [( q ; m ,and ` j q + 1 .....50 4.6.1SubmoduleStructure...........................50 5

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4.6.2 2 -ElementaryDivisorsof S and K when m IsOdd............51 4.6.3 2 -ElementaryDivisorsof S and K when m IsEven............52 4.6.4 ` -ElementaryDivisorsof S and K when m IsEven, ` , 2 and ` j q + 1 .53 4.6.5 ` -ElementaryDivisorsof K when m IsOdd, ` , 2 and ` j q + 1 ......54 4.7 ` -ElementaryDivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.571 -1.794 Td [(o + q ; m ,and ` j q + 1 .....54 4.7.1SubmoduleStructure............................55 4.7.2 2 -ElementaryDivisorsof S and K when m IsEven............56 4.7.3 2 -ElementaryDivisorsof S and K when m IsOdd............57 4.7.4 ` -ElementaryDivisorsof S and K when m isOdd,and ` j q + 1 .....58 4.7.5 ` -ElementaryDivisorsof K when m IsEvenand ` j q + 1 .........60 4.8 ` -ElementaryDivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.571 -1.794 Td [(ue q ; m ,and ` j q + 1 .....61 4.8.1SubmoduleStructure...........................61 4.8.2 ` -ElementaryDivisorsof S and K when ` j q + 1 and ` m ........61 4.8.3 ` -ElementaryDivisorsof S and K when ` j q + 1 ,and ` j m .......65 4.9 ` -ElementaryDivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.571 -1.793 Td [(uo q ; m ,and ` j q + 1 .....67 4.9.1SubmoduleStructure...........................67 4.9.2ElementaryDivisorsof S and K ,when ` m ,and ` j q + 1 .......68 4.9.3ElementaryDivisorsof S and K ,when ` j m ,and ` j q + 1 ........68 5PROOFSOFTHEOREM15ANDTHEOREM16...................70 5.1SomePropertiesof G p ;`; t ............................70 5.2CharacterSumsandBlockDiagonalFormof L ..................71 5.3TheSylow p -SubgroupoftheCriticalGroupof G p ;`; t ............75 5.4ProofofTheorem15...............................79 5.5TheCriticalGroupof G p ; 3 ; t ..........................80 REFERENCES........................................88 BIOGRAPHICALSKETCH..................................90 6

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LISTOFTABLES Table page 3-1Smithgroupof )]TJ/F71 7.9701 Tf 7.046 -1.793 Td [(s q ; m ..................................26 3-2Smithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o q ; m ..................................26 3-3Smithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o )]TJ/F68 11.9552 Tf 5.568 1.793 Td [( q ; m .................................27 3-4Smithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + q ; m .................................27 3-5Smithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(ue q ; m .................................28 3-6Smithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m .................................28 3-7Criticalgroupof )]TJ/F71 7.9701 Tf 7.045 -1.794 Td [(s q ; m .................................29 3-8Criticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o q ; m .................................29 3-9Criticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.793 Td [( q ; m ................................30 3-10Criticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + q ; m ................................30 3-11Criticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(ue q ; m .................................31 3-12Criticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m ................................31 4-1Conditionson ` ......................................36 7

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LISTOFFIGURES Figure page 2-1PetersenGraph/ KG 5 ; 2 .................................21 4-1Submodulestructure...................................43 4-2Submodulestructure...................................46 4-3Submodulestructure..................................51 4-4Submodulestructure...................................55 4-5Submodulestructure...................................62 8

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy SMITHANDCRITICALGROUPSOFGRAPHS By VenkataRaghuTejPantangi May2019 Chair:PeterK.Sin Major:Mathematics InthisDissertation,wecomputetheelementarydivisorsoftheadjacencyandLaplacian matricesoftwofamiliesofstronglyregulargraphs:PolargraphsandvanLint-Schrijver CyclotomicStronglyRegularGraphs. Polargraphshaveasverticestheisotropicone-dimensionalsubspacesofnitevector spaceswithrespecttonon-degenerateforms,withadjacencygivenbyorthogonality.Polar graphsarestronglyregulargraphsthatadmitcertainniteclassicalgroupsasautomorphisms. Let `> 2 and p beprimessuchthat p isprimitive mod ` ;alsolet t beapositive integer.Weset q : = p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t and K tobeaniteeldoforder q .By H wedenotetheunique subgroupof K withindex ` .vanLint-SchrijvergraphsaretheCayleygraphsontheadditive groupof K withconnectionset H .Thesewerediscoveredin[31]. Thestronglyregularpropertyofthesegraphsalongwiththerepresentationtheoryof niteclassicalgroupsaidusincomputingtheseelementarydivisors. 9

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CHAPTER1 INTRODUCTION Incidencestructuresareimportantobjectsincombinatorics.Anincidencestructureisa triple X ; Y ; I with X and Y beingsetsofobjectsand I X Y beingrelationbetweenthem. Wesaythat x 2 X isincidentto y 2 Y if x ; y 2 I .Anexampleofanincidencestructureis aagraph.Agraphisanincidencestructureoftheform V ; V ; I ,with I beingasymmetric relationi.e x ; y 2 I ifandonlyif y ; x 2 I . Thestructuralinformationofaniteincidencestructurecanbeencodedinaninteger matrixcalledtheincidencematrix.Therowsoftheincidentmatrix E areindexedbyelements of X anditscolumnsbyelementsof Y .For x ; y 2 X Y ,wedene E x ; y tobe 1 if x ; y 2 I andzerootherwise.Numericalalgebraicinvariantsofmatrix E becomeinvariantsforthe incidencestructure.Theincidencematrixofagraph V ; V ; I iscalledtheadjacencymatrix ofthegraphs.Theareaofspectralgraphtheorystudiesthepropertiesofagraphusingthe characteristicpolynomial,eigenvaluesandeigenvectorsofitsadjacencymatrix.Another matrixofimportanceinspectralgraphtheoryistheLaplacianmatrixwhichisdenedin2.3. EveryintegermatrixhasauniqueSmithNormalFormsee2.1.Twoincidencestructuresareisomorphicifandonlyifincidencematrixofonecanbeobtainedbypermutingrows andcolumnsoftheother.Let A and B beincidencematricesoftwoincidencestructures.It isafundamentalcombinatorialquestiontodetermineiftheyrepresentisomorphicincidence structures.Ausefulnegativecriterionisthatif A and B havedifferentSmithNormalforms, thentheydon'trepresentisomorphicincidencestructurescf.II.21of[23].ThustheSmith NormalFormsoftheadjacencyandLaplacianmatricesofagraphareimportantinvariantsof agraph. Let G = V ; V ; I beasimpleconnectedgraphwithadjacencymatrix A andLaplacian matrix L .Byabuseofnotationwemaytreat A and L asendomorphismsofthe Z V .ComputingtheSmithnormalformof A respectively L isequivalenttondingtheinvariantfactor decompositionabeliangroup Z V = A Z V respectively Z V = L Z V .Thegroup Z V = A Z V 10

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iscalledtheSmithgroupof G .Thenitepartof Z V = L Z V iscalledthe critical groupof G . Thesegroupsareimportantinvariantsofagraph.The critical groupofagraphisinoneto onecorrespondencewiththe criticalcongurations ofthechipringgamecf.2.4.1.Thusit isofsomeinteresttondtheSmithand critical groupsoffamiliesofgraphs. Smithand Critical groupsareknownforonlyasmallnumberofgraphfamilies.A particularclassofgraphsthathasprovedamenabletocomputationistheclassofstrongly regulargraphscf.2.5.Oflatetherehavebeenafewpapersthatcomputedthesegroupsfor certainfamiliesofstronglyregulargraphs,usingtherepresentationtheoryofautomorphism groupsofthesegraphse.g[10]. StatementoftheProblems Inthisthesis,wecomputetheSmithandCriticalgroupsofthefamiliesofgraphs describedbelow. 1.Let q = p t beapowerofaprime,andlet F q , F q 2 beniteeldsoforder q ,and q 2 respectively.Let V beeitheravectorspaceover F q endowedwitheitheranondegeneratealternatingbilinearformoraquadraticform,oravectorspaceover F q 2 endowedwithanon-degeneratehermitianform. Let P 0 bethesetofallsingular 1 -spacesin V .Giventwodistinct x ; y 2 P 0 ,wesay x y ifandonlyif x and y areorthogonal.Let )]TJ/F68 11.9552 Tf 6.647 0 Td [( V bethegraphon P 0 ,inwhichadjacencyis denedby .Apolargraphisagraphoftheform )]TJ/F68 11.9552 Tf 6.647 0 Td [( V . 2.Let p ;` beapairofprimessuchthat `> 2 and p isprimitive mod ` .Let t 2 Z > 0 and q = p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t .Moreoverweassumethat p q = p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 , ` )]TJ/F68 11.9552 Tf 10.811 0 Td [(1 whenever t isodd. Considertheeld K = F q andtheuniquesubgroup S of K oforder k : = q )]TJ/F68 11.9552 Tf 10.836 0 Td [(1 =` . Thenby G p ;`; t wedenotethegraphwithvertexset K andedgeset ff x ; y gj x ; y 2 K and x )]TJ/F71 11.9552 Tf 10.428 0 Td [(y 2 S g .WerefertoanysuchgraphasvanLint-Schrijvercyclotomicstrongly regulargraph. 11

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CHAPTER2 PRELIMINARIESANDDEFINITIONS 2.1SmithNormalForm Let R beaPrincipalIdealDomain, p 2 R aprime,and Z : R m ! R n bealinear transformation.BythestructuretheoremfornitelygeneratedmodulesoverPIDs,wehave f i g s i = 1 R nf 0 g suchthat i j i + 1 and coker Z R n )]TJ/F71 7.9701 Tf 5.468 0 Td [(s s M i = 1 R = i R : Let [ Z ] denotethematrixrepresentationof Z withrespecttothestandardbasis.Thenthe aboveequationtellsusthatwecannd P 2 GL n R ,and Q 2 GL m R suchthat P [ Z ] Q = 2 6 6 6 6 6 6 6 6 6 4 Y 0 s n )]TJ/F71 7.9701 Tf 5.468 0 Td [(s 0 m )]TJ/F71 7.9701 Tf 5.468 0 Td [(s s 0 n )]TJ/F71 7.9701 Tf 5.468 0 Td [(s n )]TJ/F71 7.9701 Tf 5.468 0 Td [(s 3 7 7 7 7 7 7 7 7 7 5 ; where Y = diag 1 ::: s .Thediagonalform P [ Z ] Q iscalledtheSmithnormalformof Z . Itsuniquenessuptomultiplicationof i byunitsisalsoguaranteedbytheaforementioned structuretheorem.Byinvariantfactorselementarydivisorsof Z ,wemeantheinvariant factorsrespectivelyelementarydivisorsofthemodule coker Z . Thefollowingisawellknownresultforeg.seeTheorem 2 : 4 of[29]thatgivesa descriptionoftheSmithnormalformintermsofminordeterminants. Lemma1. Let Z , [ Z ] ,and f i g 1 i s beasdescribedabove.Given 1 i s ,let d i Z betheGCDofall i i minordeterminantsof [ Z ] ,andlet d 0 Z = 1 .Wethenhave i = d i [ Z ] = d i )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 [ Z ] . Dene e j Z = jf i j v p i = j gj .Now e j Z isthemultiplicityof p j as p -elementarydivisorsofthe R -module coker Z .If R = Z ; then e j Z isthemultiplicityof p j asanelementary divisoroftheabeliangroup coker Z . Let R p bethe p -adiccompletionof R .Wehave R n p = T R m p R n )]TJ/F71 7.9701 Tf 5.467 0 Td [(s p M j > 0 R p = p j R p e j p : 12

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Dene M j Z : = f x 2 R m p j Z x 2 p j R n p g .Wehave R m = M 0 Z M 1 Z ::: M n Z . Let F = R p = pR p .If M R m p isasubmodule,dene M = M + pR m p = pR m p .Then M isan F -vectorspace.ThefollowingLemmafollowsfromthestructuretheorem. Lemma2. e j Z : = dim M j Z = M j + 1 Z . Proof. Let B ; C beorderedbasesfor R m ` and R n ` resp.,withrespecttowhich T isinSmith normalform.Let B = f 01 ; f 02 ::: f 0 e 0 ; f 11 ::: f 1 e 1 ::: f j 1 ::: f je j ::: f 1 1 ::: f 1 e 1 ; and C = g 01 ; g 02 ::: g 0 e 0 ; g 11 ::: g 1 e 1 ::: g e j 1 ::: g je j ::: ; here T f js = ` j g js ; and T f 1 s = 0 : Then B j = p j f 01 ; p j f 02 ::: p j f 0 e 0 ; p j )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 f 11 ::: p j )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 f 1 e 1 ::: f j 1 ::: f je j ::: f 1 1 ::: f 1 e 1 ; isabasis for M j : Fromthis,itisclearthat dim M j = M j + 1 = e j : Sowehave, dim M j Z )]TJ/F68 11.9552 Tf 10.26 0 Td [(dim ker Z = X t j e t Z : ThefollowingisLemma 3 : 1 of[14].Weincludeashortprooffortheconvenienceofthe reader. Lemma3. Let Z , p , M i Z ,and e i Z beasdenedabove.Let Z bethe p -adicvaluation oftheproductofacompletesetofnon-zeroinvariantfactorsof Z ,countedwithmultiplicities. Supposethatwehavetwosequencesofintegers 0 < t 1 < t 2 :::< t j and s 1 > s 2 :::> s j > s j + 1 = dim ker Z satisfyingthefollowingconditions. 1. dim M t i Z s i for 1 i j 2. Z = j P i = 1 s i )]TJ/F71 11.9552 Tf 10.858 0 Td [(s i + 1 t i , Thenthefollowinghold. a e 0 Z = m )]TJ/F71 11.9552 Tf 10.858 0 Td [(s 1 . b e t i Z = s i )]TJ/F71 11.9552 Tf 10.858 0 Td [(s i + 1 . c e a Z = 0 for a < f t 1 ::: t i ;::: t j g . Proof. Wehave Z = X i 1 ie i Z j )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 X k = 1 0 B B B B B B @ X t k i < t k + 1 ie i Z 1 C C C C C C A + X i t j ie i Z j )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 X k = 1 0 B B B B B B @ t k X t k i < t k + 1 e i Z 1 C C C C C C A + t j X i t j e i Z : 13

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Applicationofequation2givenaboveyields j )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 X k = 1 0 B B B B B B @ t k X t k i < t k + 1 e i Z 1 C C C C C C A + t j X i t j e i Z = j )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 X k = 1 t k dim M t k Z )]TJ/F68 11.9552 Tf 10.26 0 Td [(dim M t k + 1 Z + t j dim M t j Z )]TJ/F68 11.9552 Tf 10.26 0 Td [(dim ker Z : Nowapplicationofconditionsandinthestatementgivesus j )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 X k = 1 t k dim M t k Z )]TJ/F68 11.9552 Tf 10.26 0 Td [(dim M t k + 1 Z + t j dim M t j Z )]TJ/F68 11.9552 Tf 10.261 0 Td [(dim ker Z j X i = 1 s i )]TJ/F71 11.9552 Tf 10.858 0 Td [(s i + 1 t i = Z : Sotheinequations2and2areinfactequationsandthusthelemmafollows. Thefollowingresultis 12 : 8 : 4 of[9]. Lemma4. Let Z : R n ! R n bealineartransformationand 2 R beaneigenvaluefor Z ,with geometricmultiplicity c .Then dim M v p Z c . Proof. Let Fr R betheeldoffractionsof R p .Wecanextend Z toauniqueelementof End Fr R Fr R n .Forconvenience,letusdenotethiselementby Z aswell.Considerthe eigenspace V = f x 2 Fr R n j Z x = x g .Then V R n p isapure R p -submodule R p -direct summandof R n p ofrank c = dim V .Itisclearthat V R n p M d Z .As V R n p ispure,we have V R n p M v p Z . Example 1 . Let C = J )]TJ/F71 11.9552 Tf 10.746 0 Td [(I ; with J beingthe m m all 1 matrix. C haseigenvalues m )]TJ/F68 11.9552 Tf 10.448 0 Td [(1 ; )]TJ/F68 11.9552 Tf 7.603 0 Td [(1 withmultiplicities ; m )]TJ/F68 11.9552 Tf 11.418 0 Td [(1 : If diag 1 ; 2 ::: r isaSmithnormalformof C ,wehave j det C j = j Q i j = m )]TJ/F68 11.9552 Tf 10.283 0 Td [(1 : Fixaprime ` j m )]TJ/F68 11.9552 Tf 10.283 0 Td [(1 : Let e i : = e i ` : If v ` m )]TJ/F68 11.9552 Tf 10.283 0 Td [(1 = a ; as m )]TJ/F68 11.9552 Tf 10.282 0 Td [(1 isan integereigenvaluewhoseeigenspaceis 1 dimensional,by4wehave dim M a 1 . ThereforebyapplyingLemma3,setting n = 1 , t 1 = a , s 1 = 1 dim M a , s 2 = dim ker C = 0 ,wehave e a = 1 and e i = 0 for i , a . Thus Z n = C n )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 Z n Z = n )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 Z . 14

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2.2IncidenceMatrices,SmithNormalForms,andRepresentationTheory. Let X ; Y ; I beanincidencestructurewithincidencematrix E .AsdiscussedinChapter 1,thesmithnormalformof E over Z isanimportantinvariantoftheincidencestructure. Let R beaprincipalidealdomainandaprime p 2 R .Givenaset S ,by R p S wedenote the R p -freemodulewith S asabasisset.Thematrix E ,asamatrixover R p representsthe R p -linearmap : R X p ! R Y p whichisdenedby x : = P f y j x ; y 2 I g y . Let G beagroupactingtransitivelyon X and Y .Moreoverassumethatthegroup preservesincidence,thatis x ; y 2 I implies gx ; gy 2 I forall g 2 G .Then R X p and R Y p are permutationmodulesforthegroupring R p G andthe isan R p G modulehomomorphism. Givenanon-negativeinteger i ,let M i : = f m 2 R X p j m 2 p i R Y p g .As isan R p G -map, M i isan R p G -submoduleof R X p .From x 2.1,weknowthattheranksof M i 's completelydeterminethe p -elementarydivisorsof .As M i isan R p G -module,wemay userepresentationtheoryof G todeterminethesmithnormalformof E over R p . Mostoftherepresentationtheoryrequiredinthisisnicelysummarizedin[18,AppendicesA,D,E,F]. 2.3GraphsandMatrices. Agraphisanorderedpair G = V ; E comprisingofaset V ofverticestogetherwith aset E ofedges,whicharetwoelementsubsetsof V .Wesaythat a ; b 2V areadjacentif andonlyif f a ; b g2V .Thedegreeofavertex a 2V isthenumberofverticesadjacentto a . Assume V isaniteset.Fixanarbitraryorderon V .Theadjacencymatrixof G with respecttothisorderisthe jVjjVj matrix A suchthat A ij is 1 ifthe i thvertexisadjacentto the j thvertex,andzerootherwise.Thissymmetric 0 )]TJ/F68 11.9552 Tf 10.343 0 Td [(1 matrix A containsalltheinformation about G . AnotherinterestingmatrixassociatedwithagraphisitsLaplacian.TheLaplacianof G is thematrix L = D )]TJ/F71 11.9552 Tf 10.748 0 Td [(A ,where D isthediagonalmatrixwhose i thdiagonalentryisthedegree ofthe i thvertex. 15

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Linearalgebraicinvariantsofthesematricesarealsoinvariantsofthegraph.These invariantsgiveussomeinformationofthegraphstructure.Forexample,thenullityof theLaplacianmatrixgivesusthenumberofconnectedcomponentsofthegraphsee Proposition1.3.7of[9].OnemayrefertotextsonAlgebraicgraphtheorysuchas[9]and [15]formoreonhowalgebraicinvariantsofthesematricesdeterminestructuralpropertiesof thegraph. 2.4SmithandCriticalGroupsofGraphs. Let G = V ; E beaconnectedgraphwithadjacencymatrix A ,andLaplacianMatrix L . Denition5. 1.Smithgroupof G denotedby S G istheabeliangroup Z V = A Z V . 2.Criticalgroupof G denotedby K G istheabeliangroup Tor Z V = L Z V . ThesumofrowsandcolumnsoftheLaplacian L isthezerovector.Adding i throwto therstrowisaunimodulartransformation.Sothereisa P ; Q 2 Gl Z V suchthat PLQ = 2 6 6 6 6 6 6 6 6 6 4 0 n )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 n )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 L 0 3 7 7 7 7 7 7 7 7 7 5 ; where L 0 isthe ; 1 -minorof L : Thus SNF L = SNF L 0 : Considerthecharacteristicpolynomial det xI )]TJ/F71 11.9552 Tf 11.309 0 Td [(L of L : Performingrowandcolumn transformationssimilartothoseinthepreviousparagraph,weget det xI )]TJ/F71 11.9552 Tf 10.559 0 Td [(L = det P xI )]TJ/F71 11.9552 Tf 10.559 0 Td [(L = xdet 2 6 6 6 6 6 6 6 6 6 4 1 n )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 xI )]TJ/F71 11.9552 Tf 10.559 0 Td [(L 0 3 7 7 7 7 7 7 7 7 7 5 : Sotheco-efcientof x in det xI )]TJ/F71 11.9552 Tf 10.559 0 Td [(L is det 2 6 6 6 6 6 6 6 6 6 4 1 n )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F71 11.9552 Tf 7.902 0 Td [(L 0 3 7 7 7 7 7 7 7 7 7 5 = det 0 B B B B B B B B B @ 2 6 6 6 6 6 6 6 6 6 4 1 n )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F71 11.9552 Tf 7.902 0 Td [(L 0 3 7 7 7 7 7 7 7 7 7 5 Q 1 C C C C C C C C C A = det 2 6 6 6 6 6 6 6 6 6 4 j V j n )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 n )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 )]TJ/F71 11.9552 Tf 7.902 0 Td [(L 0 3 7 7 7 7 7 7 7 7 7 5 = j V j det L 0 16

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As L isasymmetricmatrix,allitseigenvaluesarereal.Let x 2 Z V ; then x T Lx = P v ; w 2 E x v )]TJ/F71 11.9552 Tf 10.922 0 Td [(x w 2 : Thisimpliesthatthenullspaceof L isthesubspacegeneratedbyall-one vector 1 = P v 2 V v : Thus L 0 isnon-singular,andas L Z V = L 0 Z V ,wehave j K G j = j det L 0 j . Fromtheaboveparagraph, jVj det L 0 istheco-efcientof x in det xI )]TJ/F71 11.9552 Tf 10.619 0 Td [(L : Sowenowhave thefollowingtheorem. Theorem6. Let G = V ; E beasimpleconnectedgraph.Thenif 1 ;::: jVj)]TJ/F68 7.9701 Tf 15.486 0 Td [(1 arethe non-zeroeigenvaluesoftheLaplacianmatrixof G ; thentheorderofthe critical group j K G j = jVj)]TJ/F68 7.9701 Tf 15.486 0 Td [(1 Q i = 1 i = jVj . ByKirchhoff'sMatrixtreetheoremforeg.seeProposition1.3.4of[9],wecanseethat the critical groupisanabeliangroupwhoseorderisthesameasthenumberofspanning treesintheunderlyinggraph.Thecriticalgroupsofvariousgraphsariseincombinatorics inthecontextofchipringgamescf.[4],astheabeliansandpilegroupinstatistical mechanicscf.[11],andalsoinarithmeticgeometry.Onemayreferto[22]foradiscussion ontheseconnections.ItisthereforeofsomeinteresttocomputetheSmithgroupsand criticalgroupsofgraphs.Weshalldiscussaboutthechipringgamein2.4.1. 2.4.1ChipFiringGame ThechipringgameongraphswasintroducedbyBjorner,Lovasz,andShorin[5].Fix agraph G onvertexset f 1 ; 2 :::; n g ,andstartbyputtingchips a i chipsatthe i thvertex.So a = a 1 ;::: a 1 ::: a n 2 Z n + ,andlet N = P a i .Eachstepofthegameinvolves ring avertex v ,thatis,onechipfrom v goestoeachofitsadjacentvertices.Avertex v canbe red ifthe numberofchipscurrentlyheldat v isatleastthedegreeof v . A position ofsize N on G isadistributionof N chipstoverticesof G ,thatisavector b = b 1 ;::: b i ;::: b n suchthat b 2 Z n + and P b i = N .A legal gameisanysequenceof positions ,suchthateach position isobtainedfromthepreviousoneby ring atavertexof G . Thefollowingwasprovedin[5]. 17

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Theorem7. Theorem2.1of[5]Givenaconnectedgraph,withaninitialdistributionof chips,eithereverylegalgamecanbecontinuedindenitely,oreverylegalgameterminated afterthesamenumberofstepsandthesamenalposition. Notethatifthegameterminates,irrespectiveoftheorderinwhichwerevertices,we endupatthesamenal position . In[4],Biggsintroducedavariantofthisgame.Inthiswexavertex q thatcanhavea negativenumberofchips,infactweneed q toalwaysbein”debt”.A conguration on G isa distributionof s j chipsateveryvertex j , q and s q = )]TJ/F39 11.9552 Tf 9.596 8.216 Td [(P s j chipsat q .Justasinthe previousgame,avertex v , q canbe red ifthenumberofchipscurrentlyheldat v isatleast thedegreeof v .Thevertex q canbe red ifnoothervertexcanbered. Eachstepofthegameinvolves ring avertex v ,thatis,onechipfrom v goestoeachof itsadjacentvertices.Avertex v , q canbe red ifthenumberofchipscurrentlyheldat v is atleastthedegreeof v . A legal gameisanysequenceof conguration ,suchthateach conguration isobtained fromthepreviousoneby ring atavertexof G .A conguration issaidtoberecurrentifitis theinitialandterminal conguration ofa legal game.A conguration issaidtobestableifno vertexbut q canbe red .A conguration issaidtobe critical ifitisbothrecurrentandstable. Let s and s 0 betheinitialandterminalpositionsofa legal game.Ifthe i thvertexis red x i times,wecanseethat s 0 )]TJ/F71 11.9552 Tf 10.858 0 Td [(s = Lx ,where L istheLaplacianof G . ThefollowingwasprovedbyBiggsin[4]. Theorem8 Biggs1997 . Anystartingcongurationofagraph G leadstoauniquecriticalconguration.Thesetofcriticalcongurationhasanaturalgroupoperationthatis isomorphictothecriticalgroup K G . ThefollowingasamplechipringgameBiggs'svariantonthe 4 -cycle. 1 2 2 -5 2 0 3 -5 0 1 3 -4 0 2 1 -3 1 0 2 -3 1 1 0 -2 18

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2 1 1 -4 0 2 1 -3 2.4.2FamiliesofGraphsWithKnown Critical Groups Itisofsomeinterestofndthe critical groupsofafamilyofgraphs. Critical grouphas beencomputedforrelativelyfewfamiliesofgraphs.Usingthefactthattheorderofthe critical groupisthenumberofspanningtrees,onecanseethatthe critical groupofanytree istrivial.Someelementarycomputationscanshowthatthe critical groupofthecomplete graphon n verticesis Z = n Z n )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 .In[4],thecriticalgroupsofthefamilyofWheelgraphs wascomputed.Thiswasdonebyndingexplicit critical congurationsthatgeneratethe critical group.In[12]thecriticalgroupsofthefamiliesofSquareRookgraphsandtheir complementswerecomputedbyndingexplicit critical congurationsthatgeneratethe critical group.The critical groupsofcompletemultipartitegraphsc.f.[17]andthoseofthe familyofhypercubegraphsc.f.[2]werecomputedbyndingtheSmithnormalformsofthe Laplacianbyexplicitunimodularmatrixoperations.Themethodologydescribedinthenext sectionwasusedtocompute critical groupsofthefollowing:PaleyGraphsc.f.[10];Peisert Graphsc.f.[26];Grassmanngraphoflinesinniteprojectivespaceandofitscomplement c.f.[7]and[14];PolarGraphsc.f.[25];andKnesergraphon2-subsetsofann-element setc.f[13]. 2.5StronglyRegularGraphs Denition9. AstronglyregulargraphSRGwithparameters v ; k ;; isa k -regulargraph on v verticessuchthat 1.anytwoadjacentverticeshave neighbours; 2.andanytwonon-adjacentverticeshave neighbours. StronglyregulargraphswereintroducedbyRajChandraBosein[6].Let )]TJ/F63 11.3574 Tf 9.804 0 Td [(beanSRG withparameters v ; k ;; : Let A beanadjacenymatrixof )]TJ/F70 11.9552 Tf 6.648 0 Td [(: Then A satises: A 2 + )]TJ/F70 11.9552 Tf 10.26 0 Td [( A + )]TJ/F71 11.9552 Tf 10.26 0 Td [(k I = J : 19

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Here J istheall-onematrix.Thiscanbeshownbyrstnotingthatthe ij thentryof A 2 isthe numberofpathsoflength 2 between i and j .Usingthedenitionofstronglyregulargraphs, wehave A 2 ij = 8 > > > > > > > > > > > < > > > > > > > > > > > : A ij if i and j areadjacent ; J )]TJ/F71 11.9552 Tf 10.559 0 Td [(A ij if i and j arenon-adjacent ; k I ij if i = j : Thuswehavetheaboveequation. Let 1 = P v 2 )]TJ/F71 11.9552 Tf 6.922 2.47 Td [(v ; thenwehave A 1 = k 1 : Let R beanyPIDand v beaunitin R : Let : R )]TJ/F36 11.9552 Tf 9.968 0 Td [(! R suchthat P v 2 )]TJ/F71 11.9552 Tf 6.424 9.4 Td [(a v v = P a v : Thenwehave R )-278(= R 1 ker : As J ker = f 0 g ; on ker ; A satises A 2 + )]TJ/F70 11.9552 Tf 10.616 0 Td [( A + )]TJ/F71 11.9552 Tf 10.616 0 Td [(k I = 0 : Soif R = C ; A hasat mostthreeeigenvalues.Let r ; s bethecomplexrootsof x 2 + )]TJ/F70 11.9552 Tf 9.361 0 Td [( x + )]TJ/F71 11.9552 Tf 9.362 0 Td [(k = 0 ; then k ; r ; s aretheeigenvaluesof A .Let ; f ; g denotethemultiplicitiesoftheseeigenvalues,thenthe characteristicequationof A is x )]TJ/F71 11.9552 Tf 10.26 0 Td [(k x )]TJ/F71 11.9552 Tf 10.26 0 Td [(r f x )]TJ/F71 11.9552 Tf 10.858 0 Td [(s g = 0 : Thematrix L = kI )]TJ/F71 11.9552 Tf 11.013 0 Td [(A istheLaplacianmatrixof )]TJ/F63 11.3574 Tf 6.647 0 Td [(.Then L hasatmostthreeeigenvalues, ; t ; u : = ; k )]TJ/F71 11.9552 Tf 10.261 0 Td [(r ; k )]TJ/F71 11.9552 Tf 10.858 0 Td [(s ofmultiplicities ; f ; g : Itfollowsthat L satises L )]TJ/F71 11.9552 Tf 10.26 0 Td [(tI l )]TJ/F71 11.9552 Tf 10.26 0 Td [(uI = )]TJ/F70 11.9552 Tf 7.603 0 Td [( J ; andthat x x )]TJ/F71 11.9552 Tf 10.26 0 Td [(tI f x )]TJ/F71 11.9552 Tf 10.26 0 Td [(uI g = 0 isthecharacteristicpolynomialof L . ThefollowingresultfollowsfromtheaboveandTheorem6. Lemma10. TheorderoftheSmithgroup S )]TJ/F68 11.9552 Tf 6.647 0 Td [( of )]TJ/F66 11.3574 Tf 9.804 0 Td [(is j kr f s g j ,andtheorderofthe critical group K )]TJ/F68 11.9552 Tf 6.647 0 Td [( is j t f u g v j . 20

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Figure2-1.PetersenGraph/ KG 5 ; 2 Example 2 . NodiscussionaboutGraphtheorycanbecompletewithoutthementionof thePetersengraph.ThePetersengraph KG 5 ; 2 isastronglyregulargraphwithparameters ; 3 ; 0 ; 1 .Fromabove,theadjacencymatrix A of KG 5 ; 2 satises A 2 + A )]TJ/F68 11.9552 Tf 10.841 0 Td [(2 I = 3 J .The eigenvaluesof A are ; )]TJ/F68 11.9552 Tf 7.603 0 Td [(2 ; 1 andtheyhavemultiplicities ; 4 ; 5 .TheLaplacian L has eigenvalues ; 5 ; 2 withmultiplicities ; 4 ; 5 .TheorderoftheSmithgroup S is 3 2 4 ,and theorderofthe critical groupis 5 4 2 5 10 . IngeneralanyKenesergraphon 2 -sets KG n ; 2 isalsostronglyregulargraph.AcomputationoftheSmithand critical groupsofthesefamiliesofgraphscanbefoundin[13]. 2.6Rank 3 PermutationGroupsandStronglyRegularGraphs. Animportantclassofstronglyregulargraphsarethosethatarisefromtherank 3 permutationgroups.Givenanatransitivegroup G ofpermutationsofaniteset ,thenumber of G a orbitsisindependentof a 2 .Thisnumberiscalledtherankofthepermutationgroup G ; .Onemayobservethatthenumberoforbitsofthenaturalactionof G on is equaltotherank. Let G ; bearank 3 permutationgroupofevenorder.Thetheactionof G on hasthreeorbitswith f v ; v j v 2 g asoneoftheorbits.Let and betheotherorbits.The graphs G = ; and G = ; ,areacomplementarypairofstronglyregulargraphs. Boththesegraphsadmit G asagroupofautomorphisms. 21

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CHAPTER3 RESULTS 3.1PolarGraphs Let q = p t beapowerofaprime,andlet F q , F q 2 beniteeldsoforder q ,and q 2 respectively.Let V beeitheravectorspaceover F q endowedwithanon-degenerate symplecticform,aquadraticform,oravectorspaceover F q 2 carryinganon-degenerate Hermitianform.By q ,wedenotethesizeoftheunderlyingeldassociatedwith V .Wenote that q = q 2 intheHermitiancaseandis q intheothercases. Let P 0 bethesetofallsingular 1 -spacesin V .Giventwodistinct v ; u 2 P 0 ,wesay v u ifandonlyif v and u areorthogonal.By G V ,wedenotethegroupofform-preserving automorphismsof V .Thentheactionof G V on P 0 isarank 3 permutationaction.The nontrivialorbitsoftheactionof G on P 0 P 0 are : = f u ; v j v u g and : = f u ; v j v u g . By )]TJ/F68 11.9552 Tf 6.647 0 Td [( V ,wedenotethestronglyregulargraph P 0 ; . Basedontheparityof dim V andthegeometryon V ,weclassify )]TJ/F68 11.9552 Tf 6.647 0 Td [( V intosixfamilies. Weassociateaparameter h 2f 0 ; 1 2 ; 1 ; 3 2 ; 2 g witheachfamily.Anotherparameterassociated witheachfamilyisthedimensionofthemaximaltotallyisotropicsubspaceof V ,denoted by z .Thegraph )]TJ/F68 11.9552 Tf 6.647 0 Td [( V isagraphon q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + h + 1 q z )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 vertices,where q isthesizeofthe underlyingeld.Anystronglyregular )]TJ/F68 11.9552 Tf 6.647 0 Td [( V isisomorphictooneofthefollowinggraphs. 1.When V = F 2 m q isasymplecticspacewith m 2 ,wedenote )]TJ/F68 11.9552 Tf 6.647 0 Td [( V by )]TJ/F71 7.9701 Tf 7.045 -1.793 Td [(s q ; m .Inthis case h = 1 , z = m ,and Sp m ; q Aut )]TJ/F71 7.9701 Tf 7.046 -1.793 Td [(s q ; m . 2.When V = F 2 m + 1 q isanendowedwithanon-degeneratequadraticformwith m 2 ,we denote )]TJ/F68 11.9552 Tf 6.647 0 Td [( V by )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o q ; m .Inthiscase h = 1 , z = m ,and O m ; q Aut )]TJ/F71 7.9701 Tf 6.648 -1.793 Td [(o q ; m . 3.When V = F 2 m q with m 3 isendowedwithanon-degenerateellipticquadratic form,wedenote )]TJ/F68 11.9552 Tf 6.647 0 Td [( V by )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o )]TJ/F68 11.9552 Tf 5.568 1.793 Td [( q ; m .Inthiscase h = 2 , z = m )]TJ/F68 11.9552 Tf 10.968 0 Td [(1 ,and O )]TJ/F68 11.9552 Tf 5.567 -4.34 Td [( m ; q Aut )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.793 Td [( q ; m . 22

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4.When V = F 2 m q with m 3 isendowedwithanon-degeneratehyperbolicquadratic form,wedenote )]TJ/F68 11.9552 Tf 6.647 0 Td [( V by )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + q ; m .Inthiscase h = 0 , z = m ,and O + m ; q Aut )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + q ; m . 5.When V = F 2 m q 2 with m 2 isendowedwithanon-degenerateHermitianquadratic form,wedenote )]TJ/F68 11.9552 Tf 6.647 0 Td [( V by )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(ue q ; m .Inthiscase h = 1 2 , z = m ,and U m ; q 2 Aut )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(ue q ; m . 6.When V = F 2 m + 1 q 2 with m 2 isendowedwithanon-degenerateHermitianquadratic form,wedenote )]TJ/F68 11.9552 Tf 6.647 0 Td [( V by )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m .Inthiscase h = 3 2 , z = m ,and U m + 1 ; q 2 Aut )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(ue q ; m . Fromnowon,weassumethat )]TJ/F68 11.9552 Tf 6.647 0 Td [( V isoneofthesixgraphsdescribedabove.A polargraph isagraphoftheform )]TJ/F68 11.9552 Tf 6.647 0 Td [( V .By G V ,wedenotethegroupofform-preservingautomorphismsof V .Forexamplewhen )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.97 -1.794 Td [(s q ; m ,wehave G V = Sp m ; q . Wedenotethenumberof j -dimensionalsubspacesof F d q ,by h d j i q .Bystandardcounting argumentswehave h d j i q = Q j i = 1 q d )]TJ/F71 5.9776 Tf 3.802 0 Td [(i + 1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 .Notethat q maybereplacedbyavariable z todene h d j i z : = Q j i = 1 z d )]TJ/F71 5.9776 Tf 3.801 0 Td [(i + 1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 . Bystandardargumentscf.[8] x 9.5,wecandeducethefollowingresult. Lemma11. Thegraph )]TJ/F68 11.9552 Tf 6.647 0 Td [( V isastronglyregulargraphwithparameters 0 B B B B B B B B B B B B B B B B @ v : = q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + h + 1 " z 1 # q ; k : = q " z )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 1 # q q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + h + 1 ; : = q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + q 2 q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + h + 1 " z )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 1 # q ; : = k q 1 C C C C C C C C C C C C C C C C A : Here q = q 2 when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V iseither )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(ue q ; m or )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m ;and q = q inothercases. Givenamatrix X 2 M n m Z ,let Ab X bethenitepartof Z n = X Z m .Nowifwe aregivenaprime ` andapositiveinteger a ,by e a wedenotethemultiplicityof ` a asan elementarydivisorofthematrix X .Then e 0 isthe ` -rankof X andfor a > 0 ,themultiplicity of ` a asanelementarydivisorof Ab X is e a ,bythestructuretheoremfornitelygenerated abeliangroups. 23

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Throughoutthethesis,theadjacencyandLaplacianmatricesof )]TJ/F68 11.9552 Tf 6.647 0 Td [( V willbedenotedby A and L .The Smithgroup of )]TJ/F68 11.9552 Tf 6.647 0 Td [( V is Ab A andthe critical groupis Ab L .Throughoutthe thesis, S and K willdenotetheSmithandcriticalgroupsof )]TJ/F68 11.9552 Tf 6.647 0 Td [( V respectively. Itfollowsfromthedenitionofstronglyregulargraphscf.[9]Theorem 8 : 1 : 2 that A satises A 2 )]TJ/F68 11.9552 Tf 10.746 0 Td [( )]TJ/F70 11.9552 Tf 10.745 0 Td [( A + )]TJ/F71 11.9552 Tf 10.745 0 Td [(k I = J .Here J isthematrixofallones.Byobservingthat 1 : = P y 2 P 0 y isaneigenvectorfor A ,correspondingtotheeigenvalue k ,wehave A )]TJ/F71 11.9552 Tf 10.558 0 Td [(kI A 2 )]TJ/F68 11.9552 Tf -449.128 -23.908 Td [( )]TJ/F70 11.9552 Tf 11.061 0 Td [( A + )]TJ/F71 11.9552 Tf 11.061 0 Td [(k I = 0 : Usingthisrelation,thefollowingLemmacanbederivedusing elementarylinearalgebra. Lemma12. Let A beanadjacencymatrixof )]TJ/F68 11.9552 Tf 6.648 0 Td [( V ,then L = kI )]TJ/F71 11.9552 Tf 10.693 0 Td [(A istheLaplacian.Let r be thepositiverootof z 2 )]TJ/F68 11.9552 Tf 10.306 0 Td [( )]TJ/F70 11.9552 Tf 10.305 0 Td [( z + )]TJ/F71 11.9552 Tf 10.305 0 Td [(k ,and s thenegativeroot.Let t = k )]TJ/F71 11.9552 Tf 10.305 0 Td [(r ,and u = k )]TJ/F71 11.9552 Tf 10.903 0 Td [(s . Thenthefollowinghold. 1.Wehave r = q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , s = )]TJ/F68 11.9552 Tf 7.603 0 Td [( q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + h + 1 , t = h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + h + 1 ,and u = q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + h + 1 h z 1 i q . 2. A hasthreeeigenvalues, k ; r ; s withmultiplicities ; f ; g .Where, f = q h q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + h + 1 q z )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 q h )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 ,and g = q q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + h + 1 q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q h )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 . 3. L haseigenvalues ; t ; u : = ; k )]TJ/F71 11.9552 Tf 10.26 0 Td [(r ; k )]TJ/F71 11.9552 Tf 10.858 0 Td [(s withmultiplicities ; f ; g . 4. z )]TJ/F71 11.9552 Tf 10.783 0 Td [(k z )]TJ/F71 11.9552 Tf 10.783 0 Td [(r z )]TJ/F71 11.9552 Tf 11.38 0 Td [(s istheminimalpolynomialof A and z z )]TJ/F71 11.9552 Tf 10.783 0 Td [(t z )]TJ/F71 11.9552 Tf 10.783 0 Td [(u istheminimal polynomialof L . 5. A )]TJ/F71 11.9552 Tf 10.26 0 Td [(rI A )]TJ/F71 11.9552 Tf 10.858 0 Td [(sI = J . 6. L )]TJ/F71 11.9552 Tf 10.26 0 Td [(tI L )]TJ/F71 11.9552 Tf 10.26 0 Td [(uI = )]TJ/F70 11.9552 Tf 7.604 0 Td [( J . 7. z )]TJ/F71 11.9552 Tf 10.26 0 Td [(kI z )]TJ/F71 11.9552 Tf 10.26 0 Td [(rI f z )]TJ/F71 11.9552 Tf 10.858 0 Td [(sI g isthecharacteristicpolynomialof A . 8. z z )]TJ/F71 11.9552 Tf 10.26 0 Td [(tI f z )]TJ/F71 11.9552 Tf 10.26 0 Td [(uI g isthecharacteristicpolynomialof L . As A isanon-singularmatrix,theorderoftheSmithgroup S = j det A j andthus, j S j = kr f s g .AsaconsequenceofKirchhoff'sMatrixTreeTheoremcf.[29],wehave j K j = t f u g v . 24

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SmithandCriticalgroupsofPolargraphs. OurmainresultsarepresentedinTheorems13and14below.APolargraphasdened intheprevioussectionisisomorphictooneof )]TJ/F71 7.9701 Tf 7.045 -1.793 Td [(s q ; m with m 2 , )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o q ; m with m 2 , )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.793 Td [( q ; m with m 3 , )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + q ; m with m 3 , )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(ue q ; m with m 2 ,and )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m with m 2 .Theorem13describestheSmithgroupsofpolargraphsandTheorem14thecritical groups.CorrespondingtothesixfamiliesofPolargraphs,thesixtablesfollowingTheorem 13respectivelyTheorem14encodethemultiplicitiesofelementarydivisorsoftheSmith resp.criticalgroupsofthesefamiliesofgraphs. Givenaprime ` ,themultiplicity e i of ` i asanelementarydivisorof A respectively L is givenintermsofparametersdenedinthersttworowsoftheTables3-1to3-6respectivelyTables3-7to3-12.Theparameters x ; f ; g denedintherstrowsofthetablesare dimensionsofcertain G V representations.Inparticular, f and g arethemultiplicitiesofthe eigenvalues r and s of A respectively.Thesecondrowsdeneparameter a ; d ; w respectively a ; b ; c ; d inthecaseof L as ` -adicvaluationsofcertaindivisorsofeigenvalues k ; r ; s of A respectively t ; u of K .Wenotethat v ` r = a + w , v ` s = d , v ` k = a + d , v ` t = a + c , and v ` u = b + d . Example 3 . The 4 thand 5 throwsoftable3-7showthatthe 2 -elementarydivisors L when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.97 -1.794 Td [(s q ; m with q odd iare 2 0 , 2 1 , 2 d + 1 , 2 d + b + 1 withmultiplicities g + 1 , f )]TJ/F71 11.9552 Tf 10.147 0 Td [(g )]TJ/F68 11.9552 Tf 10.148 0 Td [(1 , 1 ,and g )]TJ/F68 11.9552 Tf 10.147 0 Td [(1 respectively,when m iseven; iiandare 2 0 , 2 1 , 2 a + c , 2 a + c + 1 withmultiplicities g , 1 , f )]TJ/F71 11.9552 Tf 10.306 0 Td [(g )]TJ/F68 11.9552 Tf 10.306 0 Td [(1 ,and g respectively,when m isodd. Parameters a ; b ; c ; d ; f and g areasdenedinthersttworowsoftable3-7. Theorem13. Let V beaeitheravectorspaceover F q endowedwithanon-degenerate symplecticform,quadraticform,oravectorspaceover F q 2 carryinganon-degenerate Hermitianform.Furtherassume dim V 4 when V carriesasymplectic/Hermitianform, and dim V 5 when V isendowedwithanon-degeneratequadraticform.Considerthe 25

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graph )]TJ/F68 11.9552 Tf 6.647 0 Td [( V ,itsSmithgroup S andaprime ` jj S j .If ` = p ,the ` -partof S is Z = q 2 Z when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V iseither )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(ue q ; m or )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m ,and Z = q Z inothercases.If ` , p ,theelementarydivisorsof S areasdescribedinTables3-1,3-2,3-3,3-4,3-5,and3-6.Inthese, ij is 1 if i = j and 0 otherwise. f ; g : = q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q m )]TJ/F68 5.9776 Tf 3.801 0 Td [(1 + 1 2 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; q q m + 1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 2 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 a ; d ; w : = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q ; v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 ; v ` q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 Prime Arithmeticconditions Non-zerodivisormultiplicities ` = 2 m isevenand q isodd e 0 = g + 1 , e w = f )]TJ/F71 11.9552 Tf 8.023 0 Td [(g )]TJ/F68 11.9552 Tf 8.022 0 Td [(1 ,and e d + w = g + 1 . m isoddand q isodd e 0 = g , e a = 1 , e a + w = f )]TJ/F71 11.9552 Tf 11.388 0 Td [(g )]TJ/F68 11.9552 Tf 11.388 0 Td [(1 ,and e a + w + 1 = g + 1 . ` , 2 d = 0 e 0 = g + a ; 0 , e a = w ; 0 f + 1 + a ; 0 g , and e a + w = f + w ; 0 . a = w = 0 e 0 = f ,and e d = g + 1 . Table3-1.Smithgroupof )]TJ/F71 7.9701 Tf 7.046 -1.793 Td [(s q ; m . See x 4.2set h = 1 and z = m and x 4.4forcomputationoftheSmithgroupof )]TJ/F71 7.9701 Tf 7.046 -1.793 Td [(s q ; m . See x 4.2set h = 1 and z = m and x 4.5forthecomputationoftheSmithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o q ; m . x ; f ; g : = q 2 m )]TJ/F71 11.9552 Tf 10.26 0 Td [(q 2 q 2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ; q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 + 1 2 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; q q m + 1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 2 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ! a ; d ; w : = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q ; v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 ; v ` q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 Prime Arithmeticconditions Non-zerodivisormultiplicities ` = 2 m isevenand q isodd e 0 = x + 1 , e d = g )]TJ/F71 11.9552 Tf 11.769 0 Td [(x , e w = f )]TJ/F71 11.9552 Tf 11.768 0 Td [(x )]TJ/F68 11.9552 Tf 11.171 0 Td [(1 ,and e d + w = x + 1 . m isoddand q isodd e 0 = x , e 1 = g )]TJ/F71 11.9552 Tf 11.356 0 Td [(x + a ; 1 , e a = 1 + a ; 1 g )]TJ/F71 11.9552 Tf 11.356 0 Td [(x , e a + w = f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e a + w + 1 = x + 1 . ` , 2 d = 0 e 0 = g + a ; 0 , e a = w ; 0 f + 1 + a ; 0 g ,and e a + w = f + w ; 0 . a = w = 0 e 0 = f and e d = g + 1 . Table3-2.Smithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o q ; m . See x 4.2set h = 2 and z = m )]TJ/F68 11.9552 Tf 10.441 0 Td [(1 and x 4.6forcomputationoftheSmithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.794 Td [( q ; m . See x 4.2set h = 0 and z = m and x 4.7forthecomputationoftheSmithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + q ; m . See x 4.2set h = 1 = 2 and z = m and x 4.8forthecomputationoftheSmithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(ue q ; m .See x 4.2set h = 3 = 2 and z = m and x 4.9forthecomputationoftheSmithgroup of )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m . 26

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f ; g : = q 2 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q m )]TJ/F68 5.9776 Tf 3.801 0 Td [(1 + 1 q 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; q q m + 1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 a ; d ; w : = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q ; v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 ; v ` q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 Prime Arithmeticconditions Non-zerodivisormultiplicities ` = 2 m isevenand q odd. e 0 = g , e a = 1 , e a + w = f )]TJ/F71 11.9552 Tf 11.388 0 Td [(g )]TJ/F68 11.9552 Tf 11.388 0 Td [(1 ,and e a + d + w = g + 1 . m isoddand q isodd. e 0 = g + 1 , e w = f )]TJ/F71 11.9552 Tf 10.261 0 Td [(g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e w + 1 = g + 1 . ` , 2 q + 1 0mod ` and m iseven e 0 = g , e a = f )]TJ/F71 11.9552 Tf 10.26 0 Td [(g ,and e a + d = g + 1 q . )]TJ/F68 11.9552 Tf 7.603 0 Td [(1mod ` and d = 0 e 0 = g + a ; 0 , e a = w ; 0 f + 1 + a ; 0 g , and e a + w = f + w ; 0 . q . )]TJ/F68 11.9552 Tf 7.603 0 Td [(1mod ` and a = w = 0 e 0 = f ,and e d = g + 1 . Table3-3.Smithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.794 Td [( q ; m . f ; g : = q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(2 + 1 q 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; q 2 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 + 1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 a ; d ; w : = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q ; v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + 1 ; v ` q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 Prime Arithmeticconditions Non-zerodivisormultiplicities ` = 2 m isevenand q odd. e 0 = f , e 1 = g + 1 )]TJ/F71 11.9552 Tf 12.054 0 Td [(f ,and e w + 1 = f . m isoddand q isodd. e 0 = f )]TJ/F68 11.9552 Tf 11.101 0 Td [(1 , e d = g )]TJ/F71 11.9552 Tf 12.894 0 Td [(f + 1 + a ; d , e a = a ; d g )]TJ/F71 11.9552 Tf 12.054 0 Td [(f + 1 + 1 ,and e a + w + d = f . ` , 2 q + 1 0mod ` and m isodd e 0 = f )]TJ/F68 11.9552 Tf 10.421 0 Td [(1 , e a = 1 + a ; d g )]TJ/F71 11.9552 Tf 12.214 0 Td [(f + 1 , e d = a ; d + g )]TJ/F71 11.9552 Tf 12.054 0 Td [(f + 1 ,and e a + b = f . q . )]TJ/F68 11.9552 Tf 7.603 0 Td [(1mod ` and d = 0 e 0 = g + a ; 0 , e a = w ; 0 f + 1 + a ; 0 g , and e a + w = f + w ; 0 . q . )]TJ/F68 11.9552 Tf 7.603 0 Td [(1mod ` and a = w = 0 e 0 = f and e d = g + 1 . Table3-4.Smithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + q ; m . Theorem14. Let V beaeitheravectorspaceover F q endowedwithanon-degenerate symplecticform,quadraticform,oravectorspaceover F q 2 carryinganon-degenerate Hermitianform.Furtherassume dim V 4 when V carriesasymplectic/Hermitianform, and dim V 5 when V isendowedwithanon-degeneratequadraticform.Considerthe graph )]TJ/F68 11.9552 Tf 6.647 0 Td [( V ,itscriticalgroup K andaprime ` jj K j .The ` -elementarydivisorsof K are asdescribedinTables3-7,3-8,3-9,3-10,3-11,and3-12.Inthese, ij is 1 if i = j and 0 otherwise. See x 4.2set h = 1 and z = m and x 4.4forcomputationofthecriticalgroupof )]TJ/F71 6.9738 Tf 5.888 -1.495 Td [(s q ; m . See x 4.2set h = 1 and z = m and x 4.5forthecomputationofthecriticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o q ; m . See x 4.2set h = 2 and z = m )]TJ/F68 11.9552 Tf 10.412 0 Td [(1 and x 4.6forcomputationofthe critcal groupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.793 Td [( q ; m . 27

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x ; f ; g : = 0 B B B B B B B B @ q 2 m )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 q 2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ; q 2 h m 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 q + 1 ; q 3 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 1 C C C C C C C C A a ; d ; w : = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 ; v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 ; v ` q 2 )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 Prime Arithmeticconditions Non-zerodivisormultiplicities ` j q + 1 ` m and ` m )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 e 0 = x + 1 , e w = f )]TJ/F71 11.9552 Tf 11.35 0 Td [(x )]TJ/F68 11.9552 Tf 10.752 0 Td [(1 + w ; d g )]TJ/F71 11.9552 Tf 11.351 0 Td [(x , e d = w ; d f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x ,and e w + d = x + 1 . ` j m )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 e 0 = x , e a = 1 + a ; d g )]TJ/F71 11.9552 Tf 11.328 0 Td [(x , e d = g )]TJ/F71 11.9552 Tf 11.328 0 Td [(x + a ; d , e w + a = f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e w + a + d = x + 1 . ` j m e 0 = x + 1 , e d = g )]TJ/F71 11.9552 Tf 11.768 0 Td [(x , e w = f )]TJ/F71 11.9552 Tf 11.769 0 Td [(x )]TJ/F68 11.9552 Tf 11.17 0 Td [(1 ,and e w + d = x + 1 . ` q + 1 d = 0 e 0 = g + a ; 0 , e a = w ; 0 f + 1 + a ; 0 g ,and e a + w = f + w ; 0 . a = w = 0 e 0 = f and e d = g + 1 . Table3-5.Smithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(ue q ; m . f ; g : = 0 B B B B B B B B @ q 3 h m 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 q + 1 ; q 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 1 C C C C C C C C A a ; d ; w : = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 ; v ` q 2 m + 1 + 1 ; v ` q 2 )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 Prime Arithmeticconditions Non-zerodivisormultiplicities ` j q + 1 ` m e 0 = g , e a = 1 , e w + a = f )]TJ/F71 11.9552 Tf 10.551 0 Td [(g )]TJ/F68 11.9552 Tf 10.551 0 Td [(1 ,and e w + a + d = g + 1 . ` j m e 0 = g + 1 , e w = f )]TJ/F71 11.9552 Tf 10.26 0 Td [(g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e w + d = g + 1 . ` q + 1 d = 0 e 0 = g + a ; 0 , e a = w ; 0 f + 1 + a ; 0 g ,and e a + w = f + w ; 0 . a = w = 0 e 0 = f and e d = g + 1 . Table3-6.Smithgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m . See x 4.2set h = 0 and z = m and x 4.7forthecomputationofthecriticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o + q ; m . See x 4.2set h = 1 = 2 and z = m and x 4.8forthecomputationofthecriticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(ue q ; m .See x 4.2set h = 3 = 2 and z = m and x 4.9forthecomputationofthecriticalgroup of )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m . Remark. Weobservethatthetwofamiliesofpolargraphs )]TJ/F71 7.9701 Tf 7.046 -1.793 Td [(s q ; m and )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o q ; m areSRGs withthesameparametersbutdifferentSmithandcriticalgroups.Thisisanexamplewhere Smithandcriticalgroupsaredistinguishinginvariantsfortwofamiliesofisospectralgraphs. 28

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f ; g : = q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q m )]TJ/F68 5.9776 Tf 3.801 0 Td [(1 + 1 2 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; q q m + 1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 2 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 a ; b ; c ; d : = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q ; v ` h m 1 i q ; v ` q m + 1 ; v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 Prime Arithmeticconditions Non-zerodivisormultiplicities ` = 2 m isevenand q isodd e 0 = g + 1 , e 1 = f )]TJ/F71 11.9552 Tf 10.435 0 Td [(g )]TJ/F68 11.9552 Tf 10.434 0 Td [(1 , e d + 1 = 1 ,and e d + b + 1 = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . m isoddand q isodd e 0 = g , e a = 1 , e a + c = f )]TJ/F71 11.9552 Tf 11.482 0 Td [(g )]TJ/F68 11.9552 Tf 11.482 0 Td [(1 ,and e a + c + 1 = g . ` , 2 b = d = 0 e 0 = g + a ; 0 , e a = c ; 0 f )]TJ/F68 11.9552 Tf 9.623 0 Td [(1 + 1 + a ; 0 g , and e a + c = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + c ; 0 . a = c = 0 e 0 = f + d ; 0 , e d = b ; 0 g + 1 + d ; 0 f , and e b + d = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + b ; 0 Table3-7.Criticalgroupof )]TJ/F71 7.9701 Tf 7.046 -1.793 Td [(s q ; m . x ; f ; g : = q 2 m )]TJ/F71 11.9552 Tf 10.26 0 Td [(q 2 q 2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ; q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q m )]TJ/F68 5.9776 Tf 3.801 0 Td [(1 + 1 2 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; q q m + 1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 2 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ! a ; b ; c ; d : = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q ; v ` h m 1 i q ; v ` q m + 1 ; v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 Prime Arithmeticconditions Non-zerodivisormultiplicities ` = 2 m isevenand q isodd e 0 = x + 1 , e 1 = f )]TJ/F71 11.9552 Tf 8.618 0 Td [(x )]TJ/F68 11.9552 Tf 8.02 0 Td [(1 , e d + 1 = 1 + b ; 1 g )]TJ/F71 11.9552 Tf -190.224 -14.446 Td [(x , e b + d = g )]TJ/F71 11.9552 Tf 10.248 0 Td [(x + b ; 1 ,and e d + b + 1 = x )]TJ/F68 11.9552 Tf 9.651 0 Td [(1 . m isoddand q isodd e 0 = x , e 1 = g )]TJ/F71 11.9552 Tf 8.746 0 Td [(x + a ; 1 , e a = a ; 1 g )]TJ/F71 11.9552 Tf 8.746 0 Td [(x + 1 , e a + c = f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e a + c + 1 = x . ` , 2 b = d = 0 e 0 = g + a ; 0 , e a = c ; 0 f )]TJ/F68 11.9552 Tf 9.623 0 Td [(1 + 1 + a ; 0 g , and e a + c = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + c ; 0 . a = c = 0 e 0 = f + d ; 0 , e d = b ; 0 g + 1 + d ; 0 f , and e b + d = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + b ; 0 Table3-8.Criticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o q ; m . 3.2VanLint-SchrijverGraphs. Let p ;` beapairofprimeswith `> 2 and p primitivemodulo ` .Given t 2 N ,let q = p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t and k = q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ` .Let C denotethecriticalgroupof G p ;`; t .Let C p betheSylow p -subgroupof C .Let C p 0 bethelargestsubgroupof C whoseorderisnotdivisibleby p .As C isabelian,wehave C = C p C p 0 . ThefollowingtheoremdescribestheSylow p -subgroup C p ofthecriticalgroup C of G p ;`; t . Theorem15. Considerthegraph G p ;`; t with p q = p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 , ` )]TJ/F68 11.9552 Tf 10.437 0 Td [(1 whenever t isodd.Let d denote v p ` )]TJ/F68 11.9552 Tf 10.503 0 Td [(1 .Givenintegers a ; b notdivisibleby q )]TJ/F68 11.9552 Tf 10.502 0 Td [(1 ,let c a ; b denotethenumberof 29

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f ; g : = q 2 q m )]TJ/F68 5.9776 Tf 3.801 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 + 1 q 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; q q m + 1 q m )]TJ/F68 5.9776 Tf 3.801 0 Td [(2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 a ; b ; c ; d : = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q ; v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q ; v ` q m + 1 ; v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 Prime Arithmeticconditions Non-zerodivisormultiplicities ` = 2 m isevenand q odd. e 0 = g , e a = 1 , e a + 1 = f )]TJ/F71 11.9552 Tf 11.458 0 Td [(g )]TJ/F68 11.9552 Tf 11.458 0 Td [(1 ,and e a + d + 1 = g . m isoddand q isodd. e 0 = g + 1 , e c = f )]TJ/F71 11.9552 Tf 10.487 0 Td [(g )]TJ/F68 11.9552 Tf 10.487 0 Td [(1 , e c + 1 = 1 ,and e b + c + 1 = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . ` , 2 q + 1 0mod ` and m iseven e 0 = g , e a = f )]TJ/F71 11.9552 Tf 10.26 0 Td [(g ,and e a + d = g q + 1 0mod ` and m isodd e 0 = g + 1 , e c = f )]TJ/F71 11.9552 Tf 10.261 0 Td [(g ,and e b + c = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q . )]TJ/F68 11.9552 Tf 7.603 0 Td [(1mod ` and b = d = 0 e 0 = g + a ; 0 , e a = c ; 0 f )]TJ/F68 11.9552 Tf 9.623 0 Td [(1 + 1 + a ; 0 g , and e a + c = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + c ; 0 . q . )]TJ/F68 11.9552 Tf 7.604 0 Td [(1mod ` and a = c = 0 e 0 = f + d ; 0 , e d = b ; 0 g + 1 + d ; 0 f , and e b + d = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + b ; 0 Table3-9.Criticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.793 Td [( q ; m . f ; g : = q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(2 + 1 q 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; q 2 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 + 1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 a ; b ; c ; d : = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q ; v ` h m 1 i q ; v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 ; v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + 1 Prime Arithmeticconditions Non-zerodivisormultiplicities ` = 2 m isevenand q odd. e 0 = f , e c + 1 = b ; c g )]TJ/F71 11.9552 Tf 12.345 0 Td [(f + 1 + 1 , e b + 1 = g )]TJ/F71 11.9552 Tf 12.054 0 Td [(f + 1 + b ; c ,and e b + c + 1 = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 . m isoddand q isodd. e 0 = f )]TJ/F68 11.9552 Tf 10.421 0 Td [(1 , e a = 1 + a ; d g + 1 )]TJ/F71 11.9552 Tf 12.214 0 Td [(f , e d = g + 1 )]TJ/F71 11.9552 Tf 12.053 0 Td [(f + a ; d ,and e a + d + 1 = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . ` , 2 q + 1 0mod ` and m isodd e 0 = f )]TJ/F68 11.9552 Tf 10.421 0 Td [(1 , e a = 1 + a ; d g )]TJ/F71 11.9552 Tf 12.214 0 Td [(f + 1 , e d = a ; d + g )]TJ/F71 11.9552 Tf 12.054 0 Td [(f + 1 and e a + d = f )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 . q + 1 0mod ` and m iseven e 0 = f , e c = 1 + b ; c g )]TJ/F71 11.9552 Tf 13.325 0 Td [(f + 1 , e b = g )]TJ/F71 11.9552 Tf 12.054 0 Td [(f + 1 + b ; c ,and e b + c = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 . q . )]TJ/F68 11.9552 Tf 7.603 0 Td [(1mod ` and b = d = 0 e 0 = g + a ; 0 , e a = c ; 0 f )]TJ/F68 11.9552 Tf 9.623 0 Td [(1 + 1 + a ; 0 g , and e a + c = f )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 + c ; 0 . q . )]TJ/F68 11.9552 Tf 7.603 0 Td [(1mod ` and a = c = 0 e 0 = f + d ; 0 , e d = b ; 0 g + 1 + d ; 0 f , and e b + d = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + b ; 0 Table3-10.Criticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + q ; m . carrieswhenaddingthe p -adicexpansionsof a and b mod q )]TJ/F68 11.9552 Tf 10.598 0 Td [(1 .Let L betheLaplacian matrixandlet C bethecriticalgroupof G p ;`; t .For 1 i k )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,let min i = min f c i + mk ; nk j 0 m ` )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 and 0 < n ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 g : Givenanon-zeropositiveinteger j ,let e j bethemultiplicityof p j asa p -elementarydivisorof C .By e 0 wedenotethep-rankoftheLaplacian L of G p ;`; t .Thenthefollowingaretrue. 30

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x ; f ; g : = 0 B B B B B B B B @ q 2 m )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 q 2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ; q 2 h m 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 q + 1 ; q 3 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 1 C C C C C C C C A a ; b ; c ; d : = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 ; v ` h m 1 i q 2 ; v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 ; v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 Prime Arithmeticconditions Non-zerodivisormultiplicities ` j q + 1 ` m and ` m )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 e 0 = x + 1 , e d = f )]TJ/F71 11.9552 Tf 11.508 0 Td [(x )]TJ/F68 11.9552 Tf 10.911 0 Td [(1 + c ; d g )]TJ/F71 11.9552 Tf 11.508 0 Td [(x , e c = c ; d f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x ,and e c + d = x . ` j m )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 e 0 = x , e a = 1 + a ; c g )]TJ/F71 11.9552 Tf 11.068 0 Td [(x , e c = a ; c + g )]TJ/F71 11.9552 Tf 11.067 0 Td [(x , e a + d = f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e c + d = x . ` j m e 0 = x + 1 , e d = f )]TJ/F71 11.9552 Tf 11.096 0 Td [(x )]TJ/F68 11.9552 Tf 10.499 0 Td [(1 , e b + d = g )]TJ/F71 11.9552 Tf 11.096 0 Td [(x + b ; d , e 2 d = 1 + b ; d g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x ,and e b + 2 d = x )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 . ` q + 1 b = d = 0 e 0 = g + a ; 0 , e a = c ; 0 f )]TJ/F68 11.9552 Tf 10.324 0 Td [(1 + 1 + a ; 0 g ,and e a + c = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + c ; 0 . a = c = 0 e 0 = f + d ; 0 , e d = b ; 0 g + 1 + d ; 0 f ,and e b + d = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + b ; 0 Table3-11.Criticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(ue q ; m . f ; g : = 0 B B B B B B B B @ q 3 h m 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 q + 1 ; q 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 1 C C C C C C C C A a ; b ; c ; d : = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 ; v ` h m 1 i q 2 ; v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 ; v ` q 2 m + 1 + 1 Prime Arithmeticconditions Non-zerodivisormultiplicities ` j q + 1 ` m e 0 = g , e a = 1 , e a + d = f )]TJ/F71 11.9552 Tf 9.434 0 Td [(g )]TJ/F68 11.9552 Tf 9.434 0 Td [(1 ,and e a + c + d = g . ` j m e 0 = g + 1 , e d = f )]TJ/F71 11.9552 Tf 9.466 0 Td [(g )]TJ/F68 11.9552 Tf 9.465 0 Td [(1 , e 2 d = 1 , e b + 2 d = g )]TJ/F68 11.9552 Tf 9.465 0 Td [(1 ` q + 1 b = d = 0 e 0 = g + a ; 0 , e a = c ; 0 f )]TJ/F68 11.9552 Tf 10.324 0 Td [(1 + 1 + a ; 0 g ,and e a + c = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + c ; 0 . a = c = 0 e 0 = f + d ; 0 , e d = b ; 0 g + 1 + d ; 0 f ,and e b + d = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + b ; 0 Table3-12.Criticalgroupof )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m . 1. e 0 = jf i j 1 i k )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 and min i = 0 gj + 2 and e ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t + d = jf i j min i = 0 gj . 2. e j = jf i j 1 i k )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 and min i = j gj for 0 < j < ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 . 3. e j = e ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t + d )]TJ/F71 7.9701 Tf 6.265 0 Td [(j for 0 < j < ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 . 4.If p ` )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 ,then e ` )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 t 2 = q + 1 )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 P j < t e j . 5.If p j ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,then a e ` )]TJ/F68 5.9776 Tf 3.801 0 Td [(1 t 2 + d = k + 2 )]TJ/F39 11.9552 Tf 10.85 8.216 Td [(P j < t e j and b e ` )]TJ/F68 5.9776 Tf 3.801 0 Td [(1 t 2 = ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 k )]TJ/F39 11.9552 Tf 10.85 8.216 Td [(P j < t e j . 6. e j = 0 forallother j . 31

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WeprovetheaboveTheoremin x 5.3. Inthecaseof G p ; 3 ; t ,applicationofthetransfermatrixmethodcf.Section4.7of[28] leadsustoarecursivealgorithmthatoutputsclosedformexpressionsformultiplicitiesof p -elementarydivisorsof C .Asaconsequence,wealsodetermineaclosedformexpression forthe p -ranki.e e 0 inthecontextoftheTheoremaboveoftheLaplacian.Thefollowing theoremgivesaquickrecursivealgorithmtocompute p -elementarydivisors.Theproofof thefollowingresultisin x 5.5. Let P = p + 1 3 2 x 2 y 2 + x 2 y + xy 2 + x + y + 1 + p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 3 2 3 xy , R = p 2 x 3 y 3 and Q = p + 1 3 2 xy x 2 y 2 + x 2 y + xy 2 + x + y + 1 + 2 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 3 2 3 x 2 y 2 .Wedenethepolynomial C t 2 C [ x ; y ] recursivelyasfollows: C = 2 P C = 2 P 2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 Q ; C = 6 R + 2 P 3 )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 QP )]TJ/F68 11.9552 Tf 10.261 0 Td [(2 PQ ; and C t = PC t )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 )]TJ/F71 11.9552 Tf 10.559 0 Td [(QC t )]TJ/F68 11.9552 Tf 10.26 0 Td [(4 + RC t )]TJ/F68 11.9552 Tf 10.26 0 Td [(6 for t > 3 : Theorem16. Let C p betheSylow p -subgroupofthecriticalgroupofthegraph G p ; 3 ; t with p ; t , ; 1 .Givenanon-zeropositiveinteger j ,let e j bethemultiplicityof p j asa p -elementarydivisorof C .By e 0 wedenotethep-rankoftheLaplacian L of G p ; 3 ; t .Let e a ; b bethecoefcientof x a y b in C t .Thenthefollowingaretrue.Here ij istheKronecker deltafunction. 1. e 0 = e 2 t + 2 ; p + 2 = p + 1 3 2 t t + 1 )]TJ/F68 11.9552 Tf 10.261 0 Td [(2 . 2.For a < t ,wehave e a = e 2 t + 2 ; p )]TJ/F71 7.9701 Tf 5.069 0 Td [(a = P a < b t e a ; b 3. e t + 2 ; p = k + 2 )]TJ/F39 11.9552 Tf 10.849 8.216 Td [(P j < t e j + )]TJ/F70 11.9552 Tf 10.26 0 Td [( 2 ; p k )]TJ/F39 11.9552 Tf 10.85 8.216 Td [(P j < t e j . 4. e t = )]TJ/F70 11.9552 Tf 10.26 0 Td [( 2 ; p k + 2 )]TJ/F39 11.9552 Tf 10.85 8.216 Td [(P j < t e j + k )]TJ/F39 11.9552 Tf 10.85 8.216 Td [(P j < t e j . 5. e a = 0 forallother a . 32

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Let X bethecomplexcharactertableof K and A theadjacencymatrixof G p ;`; t .Then alltheentriesof X liein Z [ ] forsomeprimitive p throotofunity .Wehavebycharacter orthogonality 1 q XX t = I and 1 q XAX t = diag r ; where runsoveradditivecharactersof K and r isasdenedin x 5.1.Notethat r isan eigenvalueof A .Wenotethateveryprime m , p isunramiedin Q [ ] .Let m beaprime lyingover m ,thentherelation3showssimilarityofmatricesoverthelocalPID Z [ ] m . Wecannowconcludethat L = kI )]TJ/F71 11.9552 Tf 11.118 0 Td [(A issimilarto diag ; u ::: u | {z } k times ; v ::: v | {z } q )]TJ/F71 7.9701 Tf 5.069 0 Td [(k )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 times ,over Z [ ] m , forallprimes m , p .Thissimilarityimplies Z [ ] m -equivalenceofmatrices.Wehavenow provedthefollowingresult. Theorem17. Considerthegraph G p ;`; t with p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 , ` )]TJ/F68 11.9552 Tf 11.196 0 Td [(1 .Let C p 0 bethelargest subgroupof C whoseorderisnotdivisibleby p .Then C p 0 Z u 0 Z k Z v 0 Z q )]TJ/F71 7.9701 Tf 5.069 0 Td [(k )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 .Here v 0 is thebiggestdivisorof p q p q + )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 t + 1 ` thatiscoprimeto p ,and u 0 isthebiggestdivisorof u = v + )]TJ/F68 11.9552 Tf 7.603 0 Td [(1 t p q thatiscoprimeto p . Example 4 . ImplementingtheRecursionin3inacomputeralgebrasystemsuchas Sage,wecancompute C .NowapplicationofTheorems16and17yieldthecriticalgroups ofthefamilyofgraphs G p ; 3 ; 4 p ,with p runningoverprimesprimitive mod3 . The 2 -partofthecriticalgroupof G ; 3 ; 4 is 9 Q i = 1 Z 2 i Z ! e i ,where [ e i ] 9 i = 1 = [32 ; 8 ; 16 ; 84 ; 1 ; 16 ; 8 ; 32 ; 28] : The 2 -complementofthecriticalgroupof G ; 3 ; 4 is Z = 15 Z . TheSylow p -subgroupofthecriticalgroupof G p ; 3 ; 4 with p , 2 8 Q i = 1 Z p i Z ! e i p ,where 1. e 8 p = 510 p + 1 3 ! 8 )]TJ/F68 11.9552 Tf 10.261 0 Td [(2 , 2. e 1 p = e 7 p = 256 = 6561 p 8 + 1040 = 6561 p 7 + 1120 = 6561 p 6 )]TJ/F68 11.9552 Tf 8.928 0 Td [(784 = 6561 p 5 )]TJ/F68 11.9552 Tf 8.928 0 Td [(2240 = 6561 p 4 )]TJ/F68 11.9552 Tf -430.379 -23.908 Td [(784 = 6561 p 3 + 1120 = 6561 p 2 + 1040 = 6561 p + 256 = 6561 , 33

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3. e 2 p = e 6 p = 776 = 6561 p 8 + 592 = 6561 p 7 )]TJ/F68 11.9552 Tf 9.593 0 Td [(2248 = 6561 p 6 )]TJ/F68 11.9552 Tf 9.593 0 Td [(1904 = 6561 p 5 + 320 = 6561 p 4 )]TJ/F68 11.9552 Tf -430.379 -23.908 Td [(1904 = 6561 p 3 )]TJ/F68 11.9552 Tf 10.26 0 Td [(2248 = 6561 p 2 + 592 = 6561 p + 776 = 6561 , 4. e 3 p = e 5 p = 304 = 2187 p 8 )]TJ/F68 11.9552 Tf 10.479 0 Td [(448 = 2187 p 7 )]TJ/F68 11.9552 Tf 10.48 0 Td [(128 = 2187 p 6 + 608 = 2187 p 5 )]TJ/F68 11.9552 Tf 10.48 0 Td [(32 = 2187 p 4 + 608 = 2187 p 3 )]TJ/F68 11.9552 Tf 10.26 0 Td [(128 = 2187 p 2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(448 = 2187 p + 304 = 2187 , 5.and e 4 p = 871 = 2187 p 8 )]TJ/F68 11.9552 Tf 10.843 0 Td [(352 = 2187 p 7 + 448 = 2187 p 6 )]TJ/F68 11.9552 Tf 10.843 0 Td [(544 = 2187 p 5 )]TJ/F68 11.9552 Tf 10.842 0 Td [(56 = 2187 p 4 )]TJ/F68 11.9552 Tf -409.841 -23.908 Td [(544 = 2187 p 3 + 448 = 2187 p 2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(352 = 2187 p + 871 = 2187 . The p -complementofthecriticalgroupof G p ; 3 ; 4 with p , 2 is Z = u 0 v 0 Z ,where u 0 = p 4 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 3 and v 0 = p 4 + 2 3 . Remark. Foraxed t ,Theorem16impliesthatthemultiplicitiesofthe p -elementarydivisors oftheLaplacianof G p ; 3 ; t arepolynomialexpressionsin p ofdegree 2 t .Wewerehowever unabletoextendthetechniquesin x 5.5toprovesimilarresultsinthegeneralcase. 34

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CHAPTER4 PROOFSOFTHEOREM13ANDTHEOREM14 Theproofsinthischapterhavebeenreproducedfrommypapers“SmithandCritical groupsofPolarGraphs”cf.[25]. 4.1Nilpotenceof A and K Modulo ` . Werecallfrom x 3.1that )]TJ/F68 11.9552 Tf 6.647 0 Td [( V isanSRGwithparameters v ; k ;; speciedinLemma 11.Followingnotationsxedin x 3.1, A willdenotetheadjacencymatrixof )]TJ/F68 11.9552 Tf 6.647 0 Td [( V and L = kI )]TJ/F71 11.9552 Tf 10.927 0 Td [(A willdenotetheLaplacianmatrix.By J ,wedenotetheall-onematrixofsamesizeas A .WealsorecallfromLemma12that A haseigenvalues k , r , s ,withmultiplicities 1 , f ,and g respectively;andthat L haseigenvalues 0 , t = k )]TJ/F71 11.9552 Tf 10.422 0 Td [(r , u = k )]TJ/F71 11.9552 Tf 11.019 0 Td [(s ,withmultiplicities 1 , f ,and g respectively.Thevaluesof r , s , t , u , f ,and g arespeciedinLemma12.Wealsoobserved that j S j = kr f s g andthat j K j = t f u g v . DeducingfromLemma12that k = )]TJ/F68 11.9552 Tf 8.948 0 Td [( qs r q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,weseethat ` jj S j ifandonlyif ` j qrs . Since tu v isaninteger,weseethat ` jj K j ifandonlyif ` j tu .InthecontextofLemma4and Lemma3,itisusefultoinvestigatethe ` -adicvaluationsofeigenvalues r , s of A ;andthose ofeigenvalues t and u of L . Given X 2 M n n Z ,by X ` wedenotethereductionof X modulo ` .Thematrix X ` is nilpotentifandonlyifalleigenvaluesof X aredivisibleby ` .Nowthediscussionintheabove paragraphandenablesustomakethefollowingobservations. 1Sincethe q iscoprimetoboth r and s ,weseethat A ` isnilpotentifandonlyif ` j r and ` j s . 2 L ` isnilpotentifandonlyif ` j t and ` j u . ThefollowingLemmacompletelyclassiesallthepairs )]TJ/F68 11.9552 Tf 6.647 0 Td [( V ;` forwhich A ` or L ` is nilpotent. Lemma18. Considerthegraph )]TJ/F68 11.9552 Tf 6.647 0 Td [( V andlet X beeithertheadjacencymatrixorthe Laplacianmatrixof )]TJ/F68 11.9552 Tf 6.647 0 Td [( V .Let ` beaprimeand X ` bethereductionof X mod ` .Then conditionsfornilpotenceof X ` areencodedinTable4-1. 35

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)]TJ/F68 11.9552 Tf 6.647 0 Td [( V ` Arithmeticconditions Nilpotenceof X ` )]TJ/F71 7.9701 Tf 7.046 -1.793 Td [(s q ; m or )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o q ; m ` = 2 q isodd True ` , 2 False )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.793 Td [( q ; m ` = 2 q isodd True ` , 2 ` j q + 1 and m iseven True ` , 2 ` j q + 1 and m isodd Truefor L ` andFalsefor A ` ` , 2 ` q + 1 False )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + q ; m ` = 2 q isodd True ` , 2 ` j q + 1 and m isodd True ` , 2 ` j q + 1 and m iseven Truefor L ` andFalsefor A ` ` , 2 ` q + 1 False )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(ue q ; m or )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m ` ` j q + 1 True ` q + 1 False Anypolargraph ` = 2 q iseven False. Table4-1.Conditionson ` . Theprooffollowsbyobservingthat X ` isnilpotentifandonlyifallthreeeigenvaluesof X aredivisibleby ` . Finding ` -elementarydivisorsof S and K inthe“non-nilpotent”casesisabiteasier. Lemma12givesus A )]TJ/F71 11.9552 Tf 10.483 0 Td [(rI A )]TJ/F71 11.9552 Tf 11.081 0 Td [(sI = J and L )]TJ/F71 11.9552 Tf 10.483 0 Td [(tI L )]TJ/F71 11.9552 Tf 10.483 0 Td [(uI = )]TJ/F70 11.9552 Tf 7.604 0 Td [( J .InthiscaseLemma4 andtheequationsabovehelpusconstructtwointegersequencessatisfyingthehypothesis ofLemma3.Wewilldothesecomputationsin x 4.2. Inthe“nilpotent”case,weuserepresentationtheoryof G V ,thegroupofformpreservinglinearisomorphismsof V .Letusconsiderthecasewhen ` isa“nilpotent”prime.Wemay treat A and L aselementsof End Z ` Z ` P 0 ,where Z ` P 0 isthefree Z ` modulewith P 0 vertex setof )]TJ/F68 11.9552 Tf 6.647 0 Td [( V asabasis.Theactionon P 0 byelementsofthegroup G V preservesadjacency andthuscommuteswiththeactionsof A and L .Thisimpliesthatthe F ` P 0 subspaces M i A and M i L arealso F ` G V -submodulesofthepermutationmodule F ` P 0 .Theactionof G V on P 0 haspermutationrank 3 .Thesubmodulestructureofthepermutationmodule F ` P 0 has beendeterminedin[21],[20],[19],and[27]in cross-characteristics ,thatis,when ` q .We usetheseresultsalongwithLemma3tonishourcomputations. 36

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4.2When A ` and L ` AreNotNilpotent. Inthissectionwedealwith )]TJ/F68 11.9552 Tf 6.647 0 Td [( V andaprime ` suchthat A ` and L ` arenotnilpotent. Table4-1canbeusedtolookupallpossiblepairs V ;` suchthat A ` equivalently L ` arenot nilpotent. 4.2.1ElementaryDivisorsof S Thegraph )]TJ/F68 11.9552 Tf 6.647 0 Td [( V isoneof )]TJ/F71 7.9701 Tf 7.045 -1.794 Td [(s q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.794 Td [( q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o + q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(uo q ; m and )]TJ/F71 7.9701 Tf 6.648 -1.794 Td [(ue q ; m . FollowingthenotationinLemma12,wehave r = h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q q )]TJ/F68 11.9552 Tf 11.206 0 Td [(1 , s = )]TJ/F68 11.9552 Tf 7.603 0 Td [( q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + h + 1 , = h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + h + 1 ,and k = q .Here q = q 2 for )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m and )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m ;and q = q forother graphs. If ` jj S j ,wesawin x 4.1that A ` isnotnilpotentifandonlyif ` doesnotdivide r and s simultaneously.Assumethat ` doesnotdivide r and s simultaneously,andthat ` jj S j .As j S j = kr f s g and k = qs r q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,weseethat ` dividesexactlyoneof q , r ,and s . Inthissubsection,weidentity A ` with A and M i A with M i . 4.2.1.0.1Case1: ` j r and ` s q . . Weset v ` h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = a and v ` q )]TJ/F68 11.9552 Tf 10.479 0 Td [(1 = w .Then v ` s = 0 , v ` k = v ` = a , v ` r = a + w , and v ` j S j = a + w f + a .As ` j r ,oneof a and w isnecessarilynon-zero. ByLemma12, z )]TJ/F71 11.9552 Tf 10.28 0 Td [(k z )]TJ/F71 11.9552 Tf 10.879 0 Td [(s g z )]TJ/F71 11.9552 Tf 10.281 0 Td [(r f isthecharacteristicequationof A .Reducingmodulo ` ,weseethat z f z )]TJETq1 0 0 1 173.643 310.243 cm[]0 d 0 J 0.669 w 0 0 m 5.511 0 l SQBT/F71 11.9552 Tf 173.643 299.83 Td [(k z )]TJETq1 0 0 1 202.845 307.457 cm[]0 d 0 J 0.669 w 0 0 m 5.248 0 l SQBT/F71 11.9552 Tf 203.443 299.83 Td [(s g isthecharacteristicpolynomialof A .ByLemma12,weobserve thatminimalpolynomialof L divides z )]TJETq1 0 0 1 277.864 286.335 cm[]0 d 0 J 0.669 w 0 0 m 5.511 0 l SQBT/F71 11.9552 Tf 277.864 275.922 Td [(k z )]TJETq1 0 0 1 310.144 283.549 cm[]0 d 0 J 0.669 w 0 0 m 5.248 0 l SQBT/F71 11.9552 Tf 310.741 275.922 Td [(s z ,andthusalltheJordanblocksof L associatedwith s havesize 1 .Therefore,thegeometricmultiplicityof s asaneigenvalueof A is g .Wecannowconcludethat dimim A )]TJETq1 0 0 1 291.954 235.734 cm[]0 d 0 J 0.669 w 0 0 m 5.248 0 l SQBT/F71 11.9552 Tf 292.551 228.106 Td [(sI = f + 1 . Lemma12giveus A A )]TJ/F71 11.9552 Tf 11.35 0 Td [(sI = )]TJ/F71 11.9552 Tf 7.604 0 Td [(r A )]TJ/F71 11.9552 Tf 11.349 0 Td [(sI + J .Since a = v ` v ` r ,weseethat im A )]TJETq1 0 0 1 109.126 187.918 cm[]0 d 0 J 0.669 w 0 0 m 5.248 0 l SQBT/F71 11.9552 Tf 109.724 180.291 Td [(sI M a .Thus dim M a f + 1 . As r isaneigenvalueofvaluation a + w ,Lemma4impliesthat dim M a + w f . WeapplyLemma3toconcludethefollowing. 37

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1.Assumethat w = 0 ,then a , 0 .As A isnon-singular, ker A = f 0 g .NowbyLemma3, setting j = 1 , t 1 = a , s 1 = f + 1 dim M a , s 2 = 0 = dim ker A ,wehave e a = f + 1 , e 0 = g ,and e i = 0 forallother i . 2.Assume a = 0 ,then w , 0 .As A isnon-singular, ker A = f 0 g .NowbyLemma3,setting j = 1 , t 1 = w , s 1 = f dim M w , s 2 = 0 ,wehave e w = f , e 0 = g + 1 and e i = 0 forall other i . 3.Assume aw , 0 .As A isnon-singular, ker A = f 0 g .NowbyLemma3,setting j = 2 , t 1 = a + w ; t 2 = ws 1 = f + 1 dim M w , s 2 = f dim M w , s 3 = 0 ,wehave e a + w = 1 , e a = f ; e 0 = g ,and e i = 0 forallother i . 4.2.1.0.2Case2: ` j s and ` r q . . Set v ` s = d .As ` r ,wehave v ` r = 0 .Then v ` k = v ` = d ,and v ` j S j = dg + d . Lemma12givesus A A )]TJ/F71 11.9552 Tf 10.546 0 Td [(rI = )]TJ/F71 11.9552 Tf 8.202 0 Td [(s A )]TJ/F71 11.9552 Tf 10.546 0 Td [(rI + J .Thisshowsthat im A )]TJETq1 0 0 1 463.629 452.092 cm[]0 d 0 J 0.669 w 0 0 m 4.926 0 l SQBT/F71 11.9552 Tf 463.629 444.465 Td [(rI M d .By Lemma12, z z )]TJ/F71 11.9552 Tf 10.583 0 Td [(s g z )]TJ/F71 11.9552 Tf 9.985 0 Td [(r f isthecharacteristicpolynomialof A .Reducingmod ` ,weseethat z g + 1 z )]TJETq1 0 0 1 111.2 404.276 cm[]0 d 0 J 0.669 w 0 0 m 4.926 0 l SQBT/F71 11.9552 Tf 111.2 396.649 Td [(r f isthecharacteristicpolynomialof A .AlsoLemma12,wecandeducethat z z )]TJETq1 0 0 1 525.89 404.276 cm[]0 d 0 J 0.669 w 0 0 m 4.926 0 l SQBT/F71 11.9552 Tf 525.89 396.649 Td [(r istheminimalpolynomialof A ,andthusthegeometricmultiplicityof r asaneigenvalueof A is f .Wecannowconcludethat dimim A )]TJETq1 0 0 1 293.35 356.461 cm[]0 d 0 J 0.669 w 0 0 m 4.926 0 l SQBT/F71 11.9552 Tf 293.35 348.833 Td [(rI = g + 1 .Therefore dim M d g + 1 .Soby Lemma3,setting j = 1 , s 1 = g + 1 , s 2 = 0 ,and t 1 = d ,wehave e d = g + 1 , e 0 = f and e i = 0 forallother i . 4.2.1.0.3Case3: ` j q and ` rs . . Set v ` q = .Then v ` k = v ` j S j = a .As k isaneigenvalueofvaluation ,Lemma4 showsthat dim M 1 .ThusbyLemma3,wededucethat e = 1 , e 0 = f + g ,and e i = 0 for allother i . 4.2.2ElementaryDivisorsof K Thegraph )]TJ/F68 11.9552 Tf 6.647 0 Td [( V isoneof )]TJ/F71 7.9701 Tf 7.045 -1.794 Td [(s q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.794 Td [( q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o + q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(uo q ; m and )]TJ/F71 7.9701 Tf 6.648 -1.794 Td [(ue q ; m . FollowingthenotationinLemma12,wehave = h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + h + 1 , t = h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + h + 1 , u = q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + h + 1 h z 1 i q .and v = h z 1 i q q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + h + 1 .Here q = q 2 for )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m and )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m ;and q = q forothergraphs. 38

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If ` jj K j ,wesawin4.1that L ` isnotnilpotentifandonlyif ` doesnotdivide t and u simultaneously.Assumethat ` doesnotdivide t and u simultaneously,andthat ` jj K j .We recallthat j K j = t f u g = v . Inthissubsection,weidentity L ` with L and M i L with M i . 4.2.2.0.1Case1: ` j t and ` u . .Inthiscase, v ` t > 0 and v ` u = 0 .Weset v ` h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = a and v ` q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + h + 1 = c .Now,wehave v ` t = a + c , v ` = a ,and v ` v = c . Since j K j = t f u g = v ,wehave v ` j K j = a + c f )]TJ/F71 11.9552 Tf 11.015 0 Td [(c .As L isamatrixofnullity 1 ,wehave dim ker L = 1 .As t isaneigenvalueof ` -valuation a + c andgeometricmultiplicity f .So Lemma4impliesthat dim M a + c f . Lemma12givesus L L )]TJ/F71 11.9552 Tf 11.211 0 Td [(uI = )]TJ/F71 11.9552 Tf 7.604 0 Td [(t L )]TJ/F71 11.9552 Tf 11.211 0 Td [(uI )]TJ/F70 11.9552 Tf 11.211 0 Td [( J .So im L )]TJETq1 0 0 1 405.336 499.908 cm[]0 d 0 J 0.669 w 0 0 m 5.978 0 l SQBT/F71 11.9552 Tf 405.336 492.28 Td [(uI M a .Againby Lemma12, z z )]TJ/F71 11.9552 Tf 10.265 0 Td [(u g z )]TJ/F71 11.9552 Tf 10.265 0 Td [(t f isthecharacteristicpolynomialof L .Reducingmod ` ,weseethat z f + 1 z )]TJETq1 0 0 1 112.635 452.092 cm[]0 d 0 J 0.669 w 0 0 m 5.978 0 l SQBT/F71 11.9552 Tf 112.635 444.465 Td [(u g isthecharacteristicpolynomialof L .FromLemma12wededucethatminimal polynomialof L divides z 2 z )]TJETq1 0 0 1 220.709 428.184 cm[]0 d 0 J 0.669 w 0 0 m 5.978 0 l SQBT/F71 11.9552 Tf 220.709 420.557 Td [(u .ThusalltheJordanblocksof L associatedwith u areofsize 1 .Thereforethegeometricmultiplicityof u asaneigenvalueof L is g .Wecannowconclude that dimim L )]TJETq1 0 0 1 153.189 380.369 cm[]0 d 0 J 0.669 w 0 0 m 5.978 0 l SQBT/F71 11.9552 Tf 153.189 372.741 Td [(uI = f + 1 ,andthus dim M c f + 1 . UsingLemma3,wearriveatthefollowingconclusions. 1.Assumethat a = 0 ,then c , 0 .SobyLemma3,setting j = 1 , s 1 = f , s 2 = dim ker L = 1 and t 1 = c ,wehave e c = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e 0 = g + 1 ,and e i = 0 forallother i . 2.Assume c = 0 ,then a , 0 .SobyLemma3,setting j = 1 , s 1 = f + 1 , s 2 = dim ker L = 1 and t 1 = a ,wehave e a = f , e 0 = g ,and e i = 0 forallother i . 3.Assume ac , 0 .ByLemma3,setting j = 2 , s 1 = f + 1 , s 2 = f , s 3 = dim ker L = 1 and t 1 = c , t 2 = a + c wehave e 0 = g , e a = 1 , e a + c = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 and e i = 0 forallother i . 4.2.2.0.2Case2: ` j u and ` t . .Inthiscase, v ` u > 0 ,and v ` t = 0 .Weset v ` h z 1 i q = b and v ` q z )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + h + 1 = d .Wehave v ` u = b + d , v ` = d ,and v ` v = b . Since j K j = t f u g = v ,wehave v ` j K j = b + d g )]TJ/F71 11.9552 Tf 10.999 0 Td [(b .As L isamatrixofnullity 1 ,wehave dim ker L = 1 .Since u isaneigenvalueofvaluation d + b andgeometricmultiplicity g , Lemma4implies dim M d + b g . 39

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ByLemma12,wehave L L )]TJ/F71 11.9552 Tf 10.744 0 Td [(tI = )]TJ/F71 11.9552 Tf 7.603 0 Td [(u L )]TJ/F71 11.9552 Tf 10.743 0 Td [(tI )]TJ/F70 11.9552 Tf 10.744 0 Td [( J .So im L )]TJETq1 0 0 1 416.685 716.981 cm[]0 d 0 J 0.669 w 0 0 m 3.539 0 l SQBT/F71 11.9552 Tf 416.685 708.045 Td [(tI M d .Lemma12 tellsthat z z )]TJ/F71 11.9552 Tf 10.812 0 Td [(u g z )]TJ/F71 11.9552 Tf 10.812 0 Td [(t f isthecharacteristicpolynomialof L .Reducingmodulo ` ,wesee that z f + 1 z )]TJETq1 0 0 1 133.681 669.166 cm[]0 d 0 J 0.669 w 0 0 m 3.539 0 l SQBT/F71 11.9552 Tf 133.681 660.229 Td [(t g isthecharacteristicpolynomialof L .FromLemma12wededucethatminimal polynomialof L divides z 2 z )]TJETq1 0 0 1 220.971 645.258 cm[]0 d 0 J 0.669 w 0 0 m 3.539 0 l SQBT/F71 11.9552 Tf 220.971 636.321 Td [(t .ThusalltheJordanblocksof L associatedwith t areofsize 1 .Thereforethegeometricmultiplicityof t asaneigenvalueof L is f .Wecannowconclude dimim L )]TJETq1 0 0 1 131.088 597.442 cm[]0 d 0 J 0.669 w 0 0 m 3.539 0 l SQBT/F71 11.9552 Tf 131.088 588.506 Td [(tI = g + 1 and dim M d g + 1 . WenowapplyLemma3toconcludethefollowing. 1.Assume b = 0 ,then d , 0 .SobyLemma3,setting j = 1 , s 1 = g + 1 , s 2 = dim ker L = 1 and t 1 = d ,wehave e 0 = f , e d = g ,and e i = 0 forallother i . 2.Assume d = 0 ,then b , 0 .SobyLemma3,setting j = 1 , s 1 = g , s 2 = dim ker L = 1 and t 1 = b ,wehave e 0 = f + 1 , e b = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e i = 0 forallother i . 3.Assume bd , 0 .ThenbyLemma3,setting j = 2 , s 1 = g + 1 , s 2 = g , s 3 = dim ker L = 1 and t 1 = d , t 2 = d + b wehave e 0 = f , e d = 1 , e d + b = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 and e i = 0 forallother i . 4.3When A ` and L ` AreNilpotent. Let ` beaprimeand )]TJ/F68 11.9552 Tf 6.647 0 Td [( V beapolargraphsuchthat A ` or L ` isnilpotent.Inthiscase, weuserepresentationtheoryof G V tocomputethe ` -elementarydivisorsof S and K . Theactionof G V on )]TJ/F68 11.9552 Tf 6.647 0 Td [( V commuteswith A and L .Thusthevectorspaces M i A and M i L areinfact G V -submodulesof F ` P 0 .Werecallthattheset P 0 whichisthesetofall singular 1 -spacesin V isthevertexsetof )]TJ/F71 11.9552 Tf 6.647 0 Td [(V . Theactionof G V on P 0 isarank 3 permutationaction.When ` isnotthecharacteteristicoftheeldassociatedwiththeunderlyingvectorspace V ,thesubmodulestructureof F ` P 0 isgivenin[19],[27],[21]and[20].Weusethesubmodulestructurespresentinliteratureto determine M i A and M i L andconsequentlyndtheelementarydivisorsof S and K . Wenowdenesomesubmodulesof F ` P 0 .Thesearesomeimportantsubmodulesof F ` P 0 denedin[19],[27],[21]and[20]. 1Givenanysubspace Z of V ,wedenote [ Z ] tobethesumofallisotropiconedimensionalsubspaceof Z .Wedenote [ V ] by 1 ,henceforthknownastheall-onevector. 40

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2Consider A and L tobeelementsof End Q ` P 0 .Dene V r = ker A )]TJ/F71 11.9552 Tf 10.403 0 Td [(rI = ker L )]TJ/F71 11.9552 Tf 10.403 0 Td [(tI , and V s = ker A )]TJ/F71 11.9552 Tf 10.936 0 Td [(sI = ker L )]TJ/F71 11.9552 Tf 10.338 0 Td [(uI .Thendene V r tobethesubspace V r Z ` P 0 of F ` P 0 ,and V s tobethesubspace V s Z ` P 0 of F ` P 0 .As V r Z ` P 0 and V s Z ` P 0 arepuresubmodulesof Z ` P 0 ,wehave dim V r = dim V r = f and dim V s = dim V s = g . 3Wedene C tobethelinearsubspaceof F ` P 0 spannedby f [ W ] j W isamaximaltotallyisotropicsubspaceofV g : 4Wedene C 0 tobethelinearsubspaceof F ` P 0 spannedby f [ W ] )]TJ/F68 11.9552 Tf 10.26 0 Td [([ W 0 ] j W ; W 0 aremaximaltotallyisotropicsubspacesofV g : 5Wedene U tobe J )]TJETq1 0 0 1 226.997 479.117 cm[]0 d 0 J 0.669 w 0 0 m 7.603 0 l SQBT/F71 11.9552 Tf 227.296 468.962 Td [(A ` F ` P 0 ,where J isthematrixofall 1 0 s. 6Wedene U 0 tobethesubspacespannedby f J )]TJETq1 0 0 1 366.54 455.21 cm[]0 d 0 J 0.669 w 0 0 m 7.603 0 l SQBT/F71 11.9552 Tf 366.839 445.054 Td [(A ` v )]TJ/F68 11.9552 Tf 10.26 0 Td [( J )]TJETq1 0 0 1 432.288 455.21 cm[]0 d 0 J 0.669 w 0 0 m 7.603 0 l SQBT/F71 11.9552 Tf 432.587 445.054 Td [(A ` u j v ; u 2 P 0 g . 7Let ; bethesymmetricbilinearformon F ` P 0 with P 0 asanorthonormalbasis.If Z isasubspace,then Z ? denotestheorthogonalcomplementof Z withrespectto ; . InthefollowingLemma,wecollectsomeinclusionrelationsinvolvingthemodules denedabove, M i A 's,and M i L 's. Lemma19. Let ` beaprimeand )]TJ/F68 11.9552 Tf 6.647 0 Td [( V beoneof f )]TJ/F71 7.9701 Tf 7.046 -1.794 Td [(s q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o )]TJ/F68 11.9552 Tf 5.568 1.794 Td [( q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o + q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(uo q ; m , )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(ue q ; m g suchthat A ` or L ` isnilpotent.Alsolet q bethesizeoftheeldassociatedwith V .Thenthefollowinghold. 1.Wehave V s M v ` s A , V r M v ` r A , V s M v ` u L ,and V t M v ` t L . 2.Given = v ` h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q ,wehave C M A and C M L . 3.Given = v ` rs ,wehave U M A . 4.Given = v ` tu ,wehave U 0 M L 5.If ` j s ,thenwehave U M L .Here = v ` ts . Proof. 1.Theeigenspaceassociatedwithaneigenvalue of A isthesameasthe eigenspaceassociatedwitheigenvalue k )]TJ/F70 11.9552 Tf 11.013 0 Td [( of L .Theproofofnowfollowsfromthe proofofLemma4. 41

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2.Let W beamaximaltotallyisotropicsubspaceof V andlet v 2 P 0 .As W isan isotropicsubspace,if v W ,then v isadjacenttoeveryother 1 -spaceof W ,atotalof h z 1 i q )]TJ/F68 11.9552 Tf 9.389 0 Td [(1 . Assumethat v 1 W .Let u 2 P 0 and u W ,then v isadjacentto u ifandonlyif u isoneofthe h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 1 -dimensionalsubspacesof v ? W .Here v ? istheorthogonalcomplementof v with respecttotheformon V .Thuswehave A [ W ] = h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 1 + r [ W ] .Since L = kI )]TJ/F71 11.9552 Tf 10.836 0 Td [(A ,wealso have L [ W ] = )]TJ/F39 11.9552 Tf 7.603 11.094 Td [(h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 1 + t [ W ] .Using h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q j t ,wearriveat2. 3.FromTable4-1,weobservethatnilpotenceof A ` or L ` implies ` j q + 1 .Thereforewe have A )]TJETq1 0 0 1 117.146 551.103 cm[]0 d 0 J 0.669 w 0 0 m 17.227 0 l SQBT/F68 11.9552 Tf 117.146 540.69 Td [(1 = qJ A + J mod ` .Wenotethat im J = F ` 1 ,andthus im A + J = im J )]TJETq1 0 0 1 500.816 550.846 cm[]0 d 0 J 0.669 w 0 0 m 7.603 0 l SQBT/F71 11.9552 Tf 501.115 540.69 Td [(A = U . Using AJ = kJ ,and = k = q andLemma12,wehave A A )]TJ/F68 11.9552 Tf 10.435 0 Td [(1 = qJ )]TJ/F68 11.9552 Tf 10.436 0 Td [( r + s I = )]TJ/F71 11.9552 Tf 7.604 0 Td [(rsI .As ` j r and ` j s ,wecanconclude3. 4.Thisfollowsbyusing LJ = 0 ,Lemma12,andcalculationssimilartothoseabove. 5.Let v 2 P 0 andlet W beanymaximaltotallyisotropicsubspaceof V .FromLemma 12,wehave L L )]TJ/F68 11.9552 Tf 10.657 0 Td [( t + u I = )]TJ/F71 11.9552 Tf 7.603 0 Td [(tuI )]TJ/F70 11.9552 Tf 10.657 0 Td [( J .Fromthecomputationsabove,wehave L [ W ] = )]TJ/F39 11.9552 Tf 7.604 11.094 Td [(h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 1 + t [ W ] .Thesetwoobservationstogetherwith LJ = 0 giveus L 0 B B B B B B B B @ L v )]TJ/F68 11.9552 Tf 10.26 0 Td [( t + u v + J v )]TJ/F70 11.9552 Tf 20.677 8.094 Td [( h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q [ W ] 1 C C C C C C C C A = )]TJ/F71 11.9552 Tf 7.604 0 Td [(tuL v )]TJ/F71 11.9552 Tf 18.908 8.094 Td [(t h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q [ W ] : Lemma11andLemma12showthat h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = )]TJ/F71 11.9552 Tf 8.201 0 Td [(s .Since ` j s ,wehave L v )]TJ/F68 11.9552 Tf 10.26 0 Td [( t + u v + J v )]TJ/F70 11.9552 Tf 20.677 8.094 Td [( h z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q [ W ] L v + J v mod ` J v )]TJ/F71 11.9552 Tf 10.559 0 Td [(A v mod ` : Now4and4yield5. 4.4 2 -ElementaryDivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.97 -1.793 Td [(s q ; m . Giventhegraph )]TJ/F71 7.9701 Tf 7.045 -1.794 Td [(s q ; m andaprime ` ,table4-1showsthat A ` equivalently L ` is nilpotentifandonlyif ` = 2 and q isodd.Inthissectionwewillcomputethe 2 -elementary divisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 18.821 -1.793 Td [(s q ; m and q isodd.Weset h = 1 ,and z = m inLemma 42

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12andLemma11togettheparametersforthisgraph.Thegraph )]TJ/F71 7.9701 Tf 7.045 -1.793 Td [(s q ; m isanSRGwith parameters v = h 2 m 1 i q , k = q h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 , = h 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 ,and = h 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q . Theadjacencymatrix A haseigenvalues k ; r ; s = k ; q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 11.327 0 Td [(1 ; )]TJ/F68 11.9552 Tf 7.604 0 Td [( + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 with multiplicities ; f ; g = 1 ; q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 + 1 2 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; q q m + 1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 2 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 .SotheLaplacian L haseigenvalues ; t ; u = ; k )]TJ/F71 11.9552 Tf 10.26 0 Td [(r ; k )]TJ/F71 11.9552 Tf 10.858 0 Td [(s = 0 ; h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q + q m ; h m 1 i q + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 withmultiplicities ; f ; g . Fromnowoninthissection,wedenote )]TJ/F71 7.9701 Tf 7.046 -1.793 Td [(s q ; m by )]TJ/F71 7.9701 Tf 7.045 -1.793 Td [(s . 4.4.1SubmoduleStructure Wenowrecallfrom x 4.3thedenitionsof C , C 0 U , U 0 , V r and V s inthecontextofthe graph )]TJ/F71 7.9701 Tf 7.045 -1.793 Td [(s .Inthiscase G V = Sp m ; q .FromTheorem2.13andRemark2.15of[19],we havethefollowingresult. Theorem20. The F 2 Sp m ; q submodulestructurefor F 2 P 0 isgivenbytheHassediagrams inFigure4-1. miseven F 2 P 0 < 1 > ? C C 0 C 0 ? C ? < 1 > f 0 g misodd F 2 P 0 C < 1 > ? C 0 C 0 ? < 1 > C ? f 0 g Figure4-1.Submodulestructure. Wehave U = C 0 ? , U 0 = C ? , dim C = f + 1 , dim C 0 = f , dim C 0 ? = g + 1 ,and dim C ? = g . 4.4.2 2 -ElementaryDivisorswhen m IsEven 4.4.2.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A 2 with A .Since m iseven, h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q isodd.Thushave v 2 q )]TJ/F68 11.9552 Tf 10.764 0 Td [(1 = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 10.763 0 Td [(1 = v 2 r .Weset v 2 q )]TJ/F68 11.9552 Tf 10.764 0 Td [(1 = w 43

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and v 2 s = v 2 + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 = d .Sowehave v 2 k = d = v 2 .As j S j = kr f s g ,weobtain v 2 j S j = d + wf + dg .Lemma19impliesthat U M d + w .ByTheorem20,weconcludethat dim M d + w dim U = g + 1 .As r isaneigenvalueof 2 -valuation w andgeometricmultiplicity f ,Lemma4impliesthat dim M w f .SobyLemma3,setting j = 2 , s 1 = f , s 2 = g + 1 , s 3 = dim ker A = 0 , t 1 = w ,and t 2 = d + w ,weobtain e 0 = g + 1 , e w = f )]TJ/F71 11.9552 Tf 10.216 0 Td [(g )]TJ/F68 11.9552 Tf 10.216 0 Td [(1 , e d + w = g + 1 , and e i = 0 forallother i . 4.4.2.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L 2 with L .Inthis case h m 1 i q isevenandthus v 2 q m )]TJ/F68 11.9552 Tf 10.612 0 Td [(1 > 1 .Thisimpliesthat v 2 q m + 1 = 1 .Set v 2 h m 1 i q = b and v 2 s = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = d .Then v 2 t = 1 , v 2 u = b + d , v 2 k = v 2 = b ,and v 2 j K j = f + b + d g )]TJ/F68 11.9552 Tf 10.26 0 Td [( b + 1 as j K j = t f u g = v . Lemma19givesus U 0 M b + d + 1 and U M d + 1 .NowbyTheorem20,wecanseethat dim M b + d + 1 g and dim M d + 1 g + 1 .As t isaneigenvalueofvaluation 1 andgeometric multiplicity f ,Lemma4givesus dim M 1 f . SobyLemma3,setting j = 3 , s 1 = f , s 2 = g + 1 , s 3 = g , s 4 = dim ker L = 1 ,and t 1 = 1 , t 2 = d + 1 ,and t 3 = d + b + 1 ,weconcludethat e 0 = g + 1 , e 1 = f )]TJ/F71 11.9552 Tf 10.569 0 Td [(g )]TJ/F68 11.9552 Tf 10.569 0 Td [(1 , e d + 1 = 1 , e b + d + 1 = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e i = 0 forallother i . 4.4.3 2 -ElementaryDivisorswhen m IsOdd 4.4.3.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A 2 with A .Inthis case m isoddandthus h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q iseven,andtherefore v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 10.681 0 Td [(1 > 1 and v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = 1 . Weset v 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = a and v 2 q )]TJ/F68 11.9552 Tf 10.713 0 Td [(1 = w .Now v 2 r = a + w , v 2 s = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = 1 ,and v 2 k = v 2 = a + 1 .As j S j = kr f s g ,wehave v 2 j S j = a + w f + g + a + 1 .ByLemma19,we have C M a and U M a + w + 1 .Theorem20implies dim M a f + 1 and dim M a + w + 1 g + 1 . As r isaneigenvalueofvaluation a + w andgeometricmultiplicity f ,Lemma4impliesthat dim M a + w f . SobyLemma3,setting j = 3 , s 1 = f + 1 , s 2 = f , s 3 = g + 1 , s 4 = dim ker A = 0 , t 1 = a , t 2 = a + w ,and t 3 = a + w + 1 ,wehave e 0 = g , e a = 1 , e a + w = f )]TJ/F71 11.9552 Tf 10.51 0 Td [(g )]TJ/F68 11.9552 Tf 10.51 0 Td [(1 , e a + w + 1 = g + 1 ,and e i = 0 forallother i . 44

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4.4.3.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L 2 with L .As h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q isevenwehave v 2 )]TJ/F68 11.9552 Tf 7.603 0 Td [(1 + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 > 1 ,andthus v 2 s = v 2 + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 = 1 .Set v 2 q m + 1 = c and v 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = a .Wethenhave v 2 t = a + c , v 2 u = 1 , v 2 v = c , v 2 k = v 2 = a + 1 ,and v 2 j K j = a + c f + g )]TJ/F71 11.9552 Tf 10.26 0 Td [(c . ByLemma19,wehave C M a and U M a + c + 1 .NowapplicationofTheorem20gives us dim M a f + 1 and dim M a + c + 1 g + 1 .As t isaneigenvalueofvaluation a + c and geometricmultiplicity f ,Lemma4impliesthat dim M a + c f . SobyLemma3,setting j = 3 , s 1 = f + 1 , s 2 = f , s 3 = g + 1 , s 4 = dim ker L = 1 , t 1 = a , t 2 = a + c ,and t 3 = a + c + 1 ,wemayconcludethat e 0 = g , e a = 1 , e a + c = f )]TJ/F71 11.9552 Tf 10.28 0 Td [(g )]TJ/F68 11.9552 Tf 10.28 0 Td [(1 , e a + c + 1 = g , and e i = 0 forallother i . 4.5 2 -ElementaryDivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.571 -1.794 Td [(o q ; m . Giventhegraph )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o q ; m andaprime ` ,table4-1showsthat A ` equivalently L ` is nilpotentifandonlyif ` = 2 and q isodd.Inthissectionwecomputethe 2 -elementary divisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.69 -1.793 Td [(o q ; m .Weset h = 1 ,and z = m inLemma12andLemma 11togetparametersforthisgraph.Thegraph )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o q ; m isanSRGwithparameters v = h 2 m 1 i q , k = q h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 , = h 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 ,and = h 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q . TheAdjacencymatrix A haseigenvalues k ; r ; s = k ; q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 11.298 0 Td [(1 ; )]TJ/F68 11.9552 Tf 7.604 0 Td [( + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 with multiplicities ; f ; g = 1 ; q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 + 1 2 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; q q m + 1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 2 q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 .SotheLaplacian L haseigenvalues ; t ; u = ; k )]TJ/F71 11.9552 Tf 10.26 0 Td [(r ; k )]TJ/F71 11.9552 Tf 10.858 0 Td [(s = 0 ; h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q + q m ; h m 1 i q + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 withmultiplicities ; f ; g . Fromnowoninthissection,wedenote )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o q ; m by )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o . 4.5.1SubmoduleStructure Wenowrecallfrom x 4.3thedenitionsof C , C 0 U , U 0 , V r and V s inthecontextofthe graph )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o .Inthiscase G V = O m + 1 ; q .FromTheorem 1 : 1 ,Corollary 7 : 5 ,andLemma 7 : 6 of[27],wehavethefollowingresult. Theorem21. Themodule U 0? hasasubmodule M containing V s suchthat dim M = V s = 1 . Therelativepositionsof M , V s , C , V r , U ? , U 0 ,and U inthe F 2 O m + 1 ; q submodulelattice of U 0 ? areasintheHassediagramsingure4-2. 45

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m isodd. U 0? C U ? V r M U V s h 1 i U 0 m iseven. U 0? C U ? V r M U V s U 0 h 1 i Figure4-2.Submodulestructure. Here y : = dim V s = U 0 = q m )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q m )]TJ/F71 11.9552 Tf 10.26 0 Td [(q 2 q + 1 , d : = dim V r = U = q m + 1 q m + q 2 q + 1 )]TJ/F68 11.9552 Tf 10.595 0 Td [(1 ,and x : = dim U 0 = q 2 m )]TJ/F71 11.9552 Tf 10.26 0 Td [(q 2 q 2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , dim C = V r = 1 , dim M = V s = 1 ,and dim U = U 0 = 1 . Remark. [27]provestheabovefor m 3 .For m = 2 ,werefertoTheorem 3 : 1 of[19].We wouldliketoaddressatypographicalerrorpresentinSection 3 of[19].Thedenitionofthe submodule C of k L 2 shouldbechangedfrom h M j M 2L 2 i k to h 1 ; 2 M j M 2L 1 i k ,andthe relateddenitionof C + shouldbesimilarlycorrected. 4.5.2 2 -ElementaryDivisorswhen m IsEven 4.5.2.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A 2 with A .Since h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q isodd,wehave v 2 q )]TJ/F68 11.9552 Tf 9.905 0 Td [(1 = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 9.904 0 Td [(1 .Setting v 2 q )]TJ/F68 11.9552 Tf 9.905 0 Td [(1 = w , v 2 + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 = d ,wehave v 2 k = d = v 2 , v 2 r = w ; v 2 s = d ,and v 2 j S j = d + dg + wf . Case1:Assumethat w = 1 . Inthiscase,as v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 11.382 0 Td [(1 = w = 1 ,wehave d = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 10.288 0 Td [(1 + 2 > 1 .As r ; s areintegereigenvalueswithnon-zero 2 -valuations, Lemma19givesus V r M 1 and V s M 1 .NowbyTheorem21,wehave V r + V s = U ? M 1 andhence dim M 1 dim U ? = f + g + 1 )]TJ/F68 11.9552 Tf 10.26 0 Td [(dim U = f + g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x . ByLemma19, U M d + 1 .ByTheorem21wehave dim U = x + 1 ,andthus 46

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dim M d + 1 x + 1 . Now s isanintegereigenvalueofgeometricmultiplicity g and 2 -valuation d .Lemma19 givesus V s M d .Also U M d + 1 M d .SobyTheorem21,wehave V s + U = M M d and hence dim M d dim M = g + 1 .Wehave, 1 f + g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F71 11.9552 Tf 10.26 0 Td [(g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + d g + 1 )]TJ/F68 11.9552 Tf 10.261 0 Td [( x + 1 + d + 1 x + 1 = f + d g + 1 = v 2 j S j : SobyLemma3,setting j = 3 , s 1 = f + g )]TJ/F71 11.9552 Tf 11.179 0 Td [(x , s 2 = g + 1 , s 3 = x + 1 , s 4 = dim ker A = 0 , t 1 = 1 , t 2 = d ,and t 3 = d + 1 ,wemayconcludethat e 0 = x + 1 , e 1 = f )]TJ/F71 11.9552 Tf 11.225 0 Td [(x )]TJ/F68 11.9552 Tf 10.627 0 Td [(1 , e d = g )]TJ/F71 11.9552 Tf 11.225 0 Td [(x , e d + 1 = x + 1 ,and e i = 0 forallother i > 0 . Case2:Assumethat w > 1 . Inthiscase, d = 1 .As r ; s areintegereigenvalueswith non-zero 2 -valuations,wehavebyLemma19, V r M 1 and V s M 1 .ThusbyTheorem21, U ? M 1 .Hence dim M 1 f + g + 1 )]TJ/F71 11.9552 Tf 10.857 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . ByLemma19,wehave U M w + 1 .Thus dim M w + 1 x + 1 and.Wehave 1 f + g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F71 11.9552 Tf 12.054 0 Td [(f + w f )]TJ/F68 11.9552 Tf 10.26 0 Td [( x + 1 + w + 1 x + 1 = wf + g + 1 = v 2 j S j : SobyLemma3,setting j = 3 , s 1 = f + g )]TJ/F71 11.9552 Tf 10.952 0 Td [(x , s 2 = f , s 3 = x + 1 , s 4 = dim ker A = 0 , t 1 = 1 , t 2 = w ,and t 3 = w + 1 ,wemayconcludethat e 0 = x + 1 , e 1 = g )]TJ/F71 11.9552 Tf 10.187 0 Td [(x , e w = f )]TJ/F71 11.9552 Tf 10.186 0 Td [(x )]TJ/F68 11.9552 Tf 9.589 0 Td [(1 , e w + 1 = x + 1 , and e i = 0 forallother i > 0 . 4.5.2.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L 2 with L .Inthis case, m iseven.Since h m 1 i q iseven,wehave v 2 q m )]TJ/F68 11.9552 Tf 11.026 0 Td [(1 > 1 and v 2 q m + 1 = 1 .Weset v 2 h m 1 i q = b and v 2 s = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = d .Sowehave v 2 t = 1 , v 2 u = d + b , v 2 k = v 2 = d , and v 2 j K j = f + b + d g )]TJ/F68 11.9552 Tf 10.26 0 Td [( b + 1 . As A L mod2 ,fromSmithgroupcomputationaboveweconcludethat dim M 1 f + g )]TJ/F71 11.9552 Tf 11.776 0 Td [(x .As u isaneigenvalueofvaluation d + b ,Lemma19implies V s M d + b ,and 47

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thus dim M d + b g .ByLemma19,wehave U 0 M d + b + 1 and U M d + 1 .Therefore dim M d + b + 1 x ,byTheorem21. Since V s M d + b M d + 1 and U M d + 1 ,byTheorem21wehave M d + 1 M .Wemay nowconcludethat dim M d + 1 g + 1 . Now, 1 f + g )]TJ/F71 11.9552 Tf 10.857 0 Td [(x )]TJ/F71 11.9552 Tf 10.26 0 Td [(g )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 + d + 1 g + 1 )]TJ/F71 11.9552 Tf 10.26 0 Td [(g + d + b g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x + d + b + 1 x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 = f + d + b g )]TJ/F68 11.9552 Tf 10.26 0 Td [( b + 1 = v 2 j K j : WeapplyLemma3toconcludethefollowing. If b > 1 ,set j = 4 , s 1 = f + g )]TJ/F71 11.9552 Tf 10.953 0 Td [(x , s 2 = g + 1 , s 3 = g , s 4 = xs 5 = dim ker L = 1 , t 1 = 1 , t 2 = d + 1 , t 3 = d + b , t 4 = d + b + 1 ,thenbyLemma3,wehave e 0 = x + 1 , e 1 = f )]TJ/F71 11.9552 Tf 10.339 0 Td [(x )]TJ/F68 11.9552 Tf 9.742 0 Td [(1 , e d + 1 = 1 , e d + b = g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x , e d + b + 1 = x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e i = 0 forallother i . If b = 1 ,set j = 3 , s 1 = f + g )]TJ/F71 11.9552 Tf 11.367 0 Td [(x , s 2 = g + 1 , s 3 = x , s 4 = dim ker L = 1 , t 1 = 1 , t 2 = d + 1 , t 3 = d + b + 1 ,thenbyLemma3,wehave e 0 = x + 1 , e 1 = f )]TJ/F71 11.9552 Tf 11.547 0 Td [(x )]TJ/F68 11.9552 Tf 10.949 0 Td [(1 , e 1 + d = g + 1 )]TJ/F71 11.9552 Tf 10.858 0 Td [(x , e d + b + 1 = x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e i = 0 forallother i . 4.5.3 2 -ElementaryDivisorsof S and K ,when m IsOdd. 4.5.3.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A 2 with A .In thiscase m isodd.As m isodd, h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q isevenandthus v 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q > 0 .Set v 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = a and v 2 q )]TJ/F68 11.9552 Tf 11.252 0 Td [(1 = w .Sowehave v 2 r = a + w , v 2 s = 1 , v 2 k = v 2 = a + 1 ,and v 2 j S j = a + w f + g + a + 1 .ByLemma19, dim M a f + 1 since dim C = f + 1 byTheorem21.As s isanintegereigenvalueofmultiplicity g with 2 -valuation 1 ,Lemma19 implies V s M 1 .Since C M a M 1 aswell,Theorem21implies M 1 V s + C = U 0 ? .So dim M 1 dim U 0? = f + g + 1 )]TJ/F68 11.9552 Tf 10.688 0 Td [(dim U 0 = f + g + 1 )]TJ/F71 11.9552 Tf 11.285 0 Td [(x .As r isanintegereigenvalue ofmultiplicity f with 2 -valuation a + w ,Lemma19implies dim M a + w f .ByLemma19,we alsohave U M a + w + 1 .FromTheorem21,weconclude dim M a + w + 1 x + 1 . 48

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Wehave 1 f + g + 1 )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [( f + 1 + a f + 1 )]TJ/F71 11.9552 Tf 12.053 0 Td [(f + a + w f )]TJ/F68 11.9552 Tf 10.26 0 Td [( x + 1 + a + b + 1 x + 1 = g + a + a + w f + 1 = v 2 j S j : WemayconcludethefollowingfromLemma3. If a > 1 ,setting j = 4 , s 1 = f + g + 1 )]TJ/F71 11.9552 Tf 8.951 0 Td [(x , s 2 = f + 1 , s 3 = f , s 4 = x + 1 , s 5 = dim ker A = 0 , t 1 = 1 , t 2 = a , t 3 = a + w ,and t 4 = a + w + 1 ,byLemma3weget e 0 = x , e 1 = g )]TJ/F71 11.9552 Tf 11.235 0 Td [(x , e a = 1 , e a + w = f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e a + w + 1 = x + 1 and e i = 0 forallother i > 0 . If a = 1 ,setting j = 3 , s 1 = f + g + 1 )]TJ/F71 11.9552 Tf 10.89 0 Td [(x , s 2 = f , s 3 = x + 1 , s 4 = dim ker A = 0 , t 1 = 1 , t 2 = a + w ,and t 3 = a + w + 1 ,byLemma3, e 0 = x , e 1 = g + 1 )]TJ/F71 11.9552 Tf 11.2 0 Td [(x , e a + w = f )]TJ/F71 11.9552 Tf 11.2 0 Td [(x )]TJ/F68 11.9552 Tf 10.602 0 Td [(1 , e a + w + 1 = x + 1 ,and e i = 0 forallother i > 0 . 4.5.3.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L 2 with L .Inthis case, m isodd.As h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q iseven,wehave v 2 )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 > 1 andthus v 2 s = v 2 + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 = 1 . Set v 2 q m + 1 = c and v 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = a .Wehave v 2 t = a + c , v 2 u = 1 , v 2 v = c , v 2 k = v 2 = a + 1 ,and v 2 j K j = a + c f + g )]TJ/F71 11.9552 Tf 10.26 0 Td [(c . ByLemma19, C M a and U M a + c + 1 .Thuswehave dim M a f + 1 and dim M a + c + 1 x + 1 ,byTheorem21.As u isanintegereigenvalueofgeometricmultiplicity g with 2 -valuation 1 ,Lemma4implies M 1 V s .Since C M a M 1 ,Theorem21implies M 1 U 0 ? .Thus dim M 1 f + g + 1 )]TJ/F71 11.9552 Tf 10.978 0 Td [(x .As t isanintegereigenvalueofmultiplicity g with 2 -valuation a + b ,Lemma19implies M a + c V r .So dim M a + c f . Wehave 1 f + g + 1 )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [( f + 1 + a f + 1 )]TJ/F71 11.9552 Tf 12.053 0 Td [(f + a + c f )]TJ/F68 11.9552 Tf 10.26 0 Td [( x + 1 + a + c + 1 x + 1 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 = g + a + c f )]TJ/F71 11.9552 Tf 10.261 0 Td [(b = v 2 j K j : UsingLemma3,weconcludethefollowing. 49

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If a > 1 ,setting j = 4 , s 1 = f + g + 1 )]TJ/F71 11.9552 Tf 9.014 0 Td [(x , s 2 = f + 1 , s 3 = f , s 4 = x + 1 , s 5 = dim ker L = 1 , t 1 = 1 , t 2 = 1 , t 3 = a + c ,and t 4 = a + c + 1 ,byLemma3wehave e 0 = x , e 1 = g )]TJ/F71 11.9552 Tf 11.194 0 Td [(x , e a = 1 , e a + c = f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e a + c + 1 = x ,and e i = 0 forallother i . If a = 1 ,bysimilarargumentswecandeducethat e 0 = x , e 1 = g )]TJ/F71 11.9552 Tf 10.355 0 Td [(x + 1 , e a + c = f )]TJ/F71 11.9552 Tf 10.355 0 Td [(x )]TJ/F68 11.9552 Tf 9.758 0 Td [(1 , e a + c + 1 = x ,and e i = 0 forallother i . 4.6 ` -ElementaryDivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.571 -1.794 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.794 Td [( q ; m ,and ` j q + 1 . Giventhegraph )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.794 Td [( q ; m andaprime ` ,table4-1showsthat A ` isnilpotentifandonly ifeitheri ` = 2 and q isodd;orii ` isoddwith ` j q + 1 and m iseven.Wealsohave L ` is nilpotentifandonlyif ` j q + 1 . Inthissectionwecomputethe ` -elementarydivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 18.116 -1.793 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.793 Td [( q ; m and `; q ; m satisfythearithmeticconditionsgivenabove. Fromnowoninthissection,wedenote )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o )]TJ/F68 11.9552 Tf 5.567 1.794 Td [( q ; m by )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o )]TJ/F63 11.3574 Tf 5.567 1.794 Td [(,and ` isaprimethatmeetsthe descriptioninthepreviousparagraph.Weset h = 2 and z = m )]TJ/F68 11.9552 Tf 10.384 0 Td [(1 inLemma12andLemma 11togettheparametersforthisgraph.Thuswegetthat )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o )]TJ/F68 11.9552 Tf 5.568 1.793 Td [( V isanSRGwithparameters v = h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q q m + 1 , k = q h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 , = q h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 )]TJ/F68 11.9552 Tf 8.593 0 Td [(1 )]TJ/F71 11.9552 Tf 8.593 0 Td [(q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 ,and = h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 . Theeigenvaluesoftheadjacencymatrix A are k ; r ; s = k ; q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F68 11.9552 Tf 10.589 0 Td [(1 ; )]TJ/F68 11.9552 Tf 7.604 0 Td [( + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; with multiplicities ; f ; g = 1 ; q 2 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q m )]TJ/F68 5.9776 Tf 3.801 0 Td [(1 + 1 q 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; q q m + 1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 . SotheLaplacian L haseigenvalues ; t ; u = 0 ; h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q + q m ; h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 with multiplicities ; f ; g . 4.6.1SubmoduleStructure Wenowrecallfrom x 4.3thedenitionsof C , C 0 U , U 0 , V r and V s inthecontextofthe graph )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o )]TJ/F63 11.3574 Tf 5.567 1.793 Td [(.Inthiscase G V = O )]TJ/F68 11.9552 Tf 5.567 -4.34 Td [( m ; q .ByCorollary 8 : 5 ,Lemma 8 : 7 ,Lemma 8 : 8 ,Lemma 8 : 9 ,Corollary 8 : 10 ,andCorollary 8 : 11 of[27],wehavethefollowingresult. Theorem22. Givenaprime ` with ` j q + 1 .Therelativepositionsof C , V r , V s , U 0 , U ,and h 1 i inthe F ` O )]TJ/F68 11.9552 Tf 5.567 -4.34 Td [( m ; q submodulestructureof C areinthegure4-3.Wehave dim C = f + 1 . 50

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` , 2 , ` j q + 1 ,and m iseven. C U V r h 1 i U 0 = V s ` , 2 , ` j q + 1 ,and m isodd. C U V r U 0 = V s h 1 i ` = 2 and m iseven C V r U h 1 i U 0 = V s ` = 2 and m isodd C V r U U 0 = V s h 1 i Figure4-3.Submodulestructure 4.6.2 2 -ElementaryDivisorsof S and K when m IsOdd. 4.6.2.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A 2 with A .Since m isodd, h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q isanoddnumber.Sowehave v 2 r = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F68 11.9552 Tf 10.859 0 Td [(1 = v 2 q )]TJ/F68 11.9552 Tf 10.859 0 Td [(1 .As h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q isanevennumber, v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 11.131 0 Td [(1 > 1 ,andthus v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = 1 = v 2 s .Wealsohave v 2 k = v 2 = 1 .Setting v 2 q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 = w ,wehave v 2 r = w and v 2 j S j = wf + g + 1 . ByLemma19,wehave U M w + 1 .SobyTheorem22,weget dim M w + 1 g + 1 .Since r isanintegereigenvaluewithvaluation w ,Lemma4implies dim M w f . Nowwehave w f )]TJ/F68 11.9552 Tf 10.516 0 Td [( g + 1 + w + 1 g + 1 = wf + g + 1 = v 2 j S j .SobyLemma3, setting j = 2 , t 1 = w , t 2 = w + 1 , s 1 = f , s 2 = g + 1 , s 3 = dim ker A = 0 ,weconcludethat e 0 = g + 1 , e w = f )]TJ/F71 11.9552 Tf 10.261 0 Td [(g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e w + 1 = g + 1 ,and e i = 0 forallother i . 51

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4.6.2.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L 2 with L .Set v 2 q m + 1 = c and v 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = b .Since m isodd,wehave v 2 s = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = 1 .So v 2 t = c , v 2 u = b + 1 , v 2 v = c + b , v 2 = 1 ,and v 2 j K j = cf + b + 1 g )]TJ/F68 11.9552 Tf 10.26 0 Td [( c + b . As t isaneigenvalueofvaluation f ,byLemma4,wehave dim M c f . Lemma19impliesthat U M c + 1 and U 0 M b + c + 1 .Therefore dim M b + c + 1 g and dim M c + 1 g + 1 ,byTheorem22. Now, c f )]TJ/F68 11.9552 Tf 10.334 0 Td [( g + 1 + c + 1 g + 1 )]TJ/F71 11.9552 Tf 10.334 0 Td [(g + b + c + 1 g )]TJ/F68 11.9552 Tf 10.334 0 Td [(1 = cf + b + 1 g )]TJ/F71 11.9552 Tf 10.333 0 Td [(b )]TJ/F71 11.9552 Tf 10.334 0 Td [(c .Soby Lemma3,setting j = 3 , s 1 = f , s 2 = g + 1 , s 3 = g , s 4 = dim ker L = 1 , t 1 = c , t 2 = c + 1 ,and t 3 = b + c + 1 ,wemayconcludethat e 0 = g + 1 , e c = f )]TJ/F71 11.9552 Tf 10.535 0 Td [(g )]TJ/F68 11.9552 Tf 10.535 0 Td [(1 , e c + 1 = 1 , e b + c + 1 = g )]TJ/F68 11.9552 Tf 10.535 0 Td [(1 and e i = 0 forallother i . 4.6.3 2 -ElementaryDivisorsof S and K when m IsEven. 4.6.3.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A 2 with A .As m iseven, h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q isevenand h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q isodd.Set v 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q = a , v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 10.947 0 Td [(1 = v 2 q )]TJ/F68 11.9552 Tf 10.947 0 Td [(1 = w and v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = d .Then v 2 r = a + w , v 2 s = d , v 2 k = v 2 = a + d ,and v 2 j S j = a + w f + dg + a + d . Lemma19,impliesthat C M a .ThusbyTheorem22wehave dim M a f + 1 . As r isaneigenvalueofvaluation a + w ,Lemma4implies dim M a + w f .ByLemma 19, U M a + d + w .ThusbyTheorem22, dim M a + d + w g + 1 . Wehave a f + 1 )]TJ/F71 11.9552 Tf 12.151 0 Td [(f + a + w f )]TJ/F68 11.9552 Tf 10.358 0 Td [( g + 1 + a + d + w g + 1 = a + w f + dg + a + d . SobyLemma3,setting j = 3 , s 1 = f + 1 , s 2 = f , s 3 = g + 1 , s 4 = dim ker A = 0 , t 1 = a , t 2 = a + w ,and t 3 = a + d + w ,wehave e 0 = g , e a = 1 ; e a + w = f )]TJ/F71 11.9552 Tf 9.692 0 Td [(g )]TJ/F68 11.9552 Tf 9.692 0 Td [(1 ; e a + d + w = g + 1 and e i = 0 forallother i . 4.6.3.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L 2 with L .As m is even, h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q isevenand h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q isodd.Set v 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q = a and v 2 s = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = d .As h m 1 i q iseven,wehave v 2 q m )]TJ/F68 11.9552 Tf 10.142 0 Td [(1 > 1 and v 2 q m + 1 = 1 .Since h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q isodd,wehave v 2 v = 1 .We have v 2 t = a + 1 , v 2 u = d ,and v 2 = a + d ,and v 2 j K j = a + 1 f + dg )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . 52

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ByLemma19,wehave C M a and U M a + d + 1 .ThereforebyTheorem22,wehave dim M a f + 1 and dim M a + d + 1 g + 1 .As t isanintegereigenvalueof L withvaluation a + 1 ,Lemma4implies dim M a + 1 f . Wehave a f + 1 )]TJ/F71 11.9552 Tf 10.886 0 Td [(f + a + 1 f )]TJ/F68 11.9552 Tf 9.092 0 Td [( g + 1 + a + d + 1 g + 1 )]TJ/F68 11.9552 Tf 9.093 0 Td [(1 = a + 1 f + dg )]TJ/F68 11.9552 Tf 9.093 0 Td [(1 = v 2 j K j . ThereforebyLemma3,setting j = 3 , s 1 = f + 1 , s 2 = f , s 3 = g + 1 , s 4 = dim ker L = 1 , t 1 = a , t 2 = a + 1 ,and t 3 = a + d + 1 ,wehave e 0 = g , e a = 1 , e a + 1 = f )]TJ/F71 11.9552 Tf 10.384 0 Td [(g )]TJ/F68 11.9552 Tf 10.384 0 Td [(1 , e a + d + 1 = g ,and e i = 0 forallother i , 0 . 4.6.4 ` -ElementaryDivisorsof S and K when m IsEven, ` , 2 and ` j q + 1 . 4.6.4.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A ` with A .Inthis case ` isanoddprimedividing q + 1 and m iseven.Thus v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q = v 2 r .Set v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q = a and v ` s = v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = d .Then v p r = a , v ` k = a + d = v ` ,and v ` j S j = af + dg + a + d . ByLemma19,wehave C M a and U M a + d .Thus dim M a f + 1 and dim M a + d g + 1 ,byTheorem22. Wehave a f + 1 )]TJ/F68 11.9552 Tf 10.792 0 Td [( g + 1 + a + d g + 1 = v ` j S j .SobyLemma3,setting j = 2 , s 1 = f + 1 , s 2 = g + 1 , s 3 = dim ker A = 0 , t 1 = a , t 2 = a + d ,weconclude e 0 = ge a = f )]TJ/F71 11.9552 Tf 10.356 0 Td [(g , e a + d = g + 1 ,and e i = 0 forallother i . 4.6.4.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L ` with L .In thiscase ` isanoddprimewith m even.Weset v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q = a ,and v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = d .As q )]TJ/F68 11.9552 Tf 19.956 0 Td [(1mod ` ,wehave v ` v = v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q q m + 1 = 0 .Thus v ` t = a , v ` u = d ,and v ` j K j = af + dg . Lemma19givesus C M a and U M a + d .NowTheorem22givesus dim M a f + 1 and dim M a + d g + 1 . Wehave a f + 1 )]TJ/F68 11.9552 Tf 9.789 0 Td [( g + 1 + a + d g + 1 )]TJ/F68 11.9552 Tf 9.789 0 Td [(1 = af + dg = .SobyLemma3,setting j = 2 , s 1 = f + 1 , s 2 = g + 1 , s 3 = dim ker A = 1 , t 1 = a , t 2 = a + d ,wehave e 0 = g , e a = f )]TJ/F71 11.9552 Tf 10.575 0 Td [(g , e a + d = g ,and e i = 0 forallother i , 0 . 53

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4.6.5 ` -ElementaryDivisorsof K when m IsOdd, ` , 2 and ` j q + 1 . Inthiscasewehave q )]TJ/F68 11.9552 Tf 20.005 0 Td [(1mod ` andthus r s )]TJ/F68 11.9552 Tf 20.005 0 Td [(2mod ` andthus ` j S j . Howeverweseethat ` j t and ` j u andthus ` jj K j .Inthissectionwecomputethe ` -elementarydivisorsof K . 4.6.5.0.1Elementarydivisorsof K . .Weidentify M i L with M i and L ` with L .Set v ` q m + 1 = c and v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = b .Since m isodd,wehave v ` s = v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = 0 .So v ` t = c , v ` u = b , v ` v = c + b , v ` = 0 ,and v ` j K j = cf + bg )]TJ/F68 11.9552 Tf 10.26 0 Td [( c + b . Lemma19givesus U 0 M b + c and V r M c .Therefore dim M b + c g and dim M c f , byTheorem22.Nowwehave v ` j K j = cf + bg )]TJ/F71 11.9552 Tf 10.26 0 Td [(c )]TJ/F71 11.9552 Tf 10.26 0 Td [(b = c f )]TJ/F71 11.9552 Tf 10.26 0 Td [(g + b + c g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . SobyLemma3,setting j = 2 , s 1 = f , s 2 = g + 1 , s 3 = dim ker L = 1 , t 1 = c , t 2 = b + c , wemayconcludethat e 0 = g + 1 , e c = f )]TJ/F71 11.9552 Tf 10.26 0 Td [(g , e b + c = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e i = 0 forallother i . 4.7 ` -ElementaryDivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.572 -1.794 Td [(o + q ; m ,and ` j q + 1 . Giventhegraph )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(o + q ; m andaprime ` ,table4-1showsthat A ` isnilpotentifandonly ifeitheri ` = 2 and q isodd;iior ` isoddwith ` j q + 1 and m isodd.Also L ` isnilpotentif andonlyif ` j q + 1 Inthissectionwecomputethe ` -elementarydivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.572 -1.793 Td [(o + q ; m and `; q ; m satisfythearithmeticconditionsgivenabove. Fromnowoninthissection,wedenote )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + q ; m by )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + ,and ` isaprimethatmeetsthe descriptioninthepreviousparagraph.Weset h = 0 ,and z = m inLemma12andLemma11 togetparametersforthisgraph.Thus )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + V isanSRGwithparameters v = h m 1 i q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 , k = q h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + 1 , = q h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + 1 )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 )]TJ/F71 11.9552 Tf 10.26 0 Td [(q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 ,and = h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + 1 . Sotheeigenvaluesoftheadjacencymatrix A are k ; r ; s = k ; q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 10.285 0 Td [(1 ; )]TJ/F68 11.9552 Tf 7.604 0 Td [( + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 ,with multiplicities ; f ; g = 1 ; q q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(2 + 1 q 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; q 2 q m )]TJ/F68 5.9776 Tf 3.801 0 Td [(1 + 1 q m )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 . So L haseigenvalues ; t ; u = ; k )]TJ/F71 11.9552 Tf 10.451 0 Td [(r ; k )]TJ/F71 11.9552 Tf 11.049 0 Td [(s = 0 ; h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; h m 1 i q + q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 with multiplicities ; f ; g . 54

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4.7.1SubmoduleStructure. Wenowrecallfrom x 4.3thedenitionsof C , C 0 U , U 0 , V r and V s inthecontextofthe graph )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(o + .Inthiscase G V = O + m ; q .ByCorollary2.10,6.5,Lemma6.6of[27]wehave thefollowingresult. Theorem23. Let ` beaprimewith ` j q + 1 .Thentherelativepositionsof U ? , C , V s , V r , U , U 0 ,and h 1 i inthe F ` O + m ; q submodulestructureof U 0 ? aregiveninthegure4-4.We have dim U 0 ? = g + 2 . ` = 2 and m iseven U 0 ? C U ? U = V r V s U 0 h 1 i ` = 2 and m isodd U 0 ? C U ? U = V r V s h 1 i U 0 ` , 2 , m isoddand ` j q + 1 U 0 ? U ? C V s U V r h 1 i U 0 ` , 2 , m isoddand ` j q + 1 U 0 ? U ? C V s U V r U 0 h 1 i Figure4-4.Submodulestructure. 55

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Here, dim U 0 = f )]TJ/F68 11.9552 Tf 9.483 0 Td [(1 , dim U = f , dim C = f + 1 , dim U ? = f + g + 1 )]TJ/F68 11.9552 Tf 9.483 0 Td [(dim U = g + 1 . 4.7.2 2 -ElementaryDivisorsof S and K when m IsEven. 4.7.2.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A 2 with A .In thiscase m iseven.Therefore h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q isodd.Sowehave v 2 r = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 10.834 0 Td [(1 = v 2 q )]TJ/F68 11.9552 Tf 10.833 0 Td [(1 , v 2 s = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + 1 = 1 ,and v 2 k = v 2 = 1 .Setting v 2 q )]TJ/F68 11.9552 Tf 7.705 0 Td [(1 = w ,wehave v 2 j S j = wf + g + 1 . Lemma4implies V r M 1 and V s M 1 .ThusbyTheorem23,weseethat M 1 U ? , andhence dim M 1 g + 1 .Lemma19givesus U M w + 1 .ThusbyTheorem23,weget dim M w + 1 f . Wehave 1 g + 1 )]TJ/F71 11.9552 Tf 12.483 0 Td [(f + w + 1 f = v 2 j S j .SobyLemma3,setting j = 2 , s 1 = g + 1 , s 2 = f , s 3 = dim ker A = 0 , t 1 = 1 ,and t 2 = w + 1 ,weconclude e 0 = f , e 1 = g + 1 )]TJ/F71 11.9552 Tf 12.517 0 Td [(f , e w + 1 = f ,and e i = 0 forall i . 4.7.2.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L 2 with L .In thiscase m iseven,sowehave v 2 s = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + 1 = 1 and v 2 k = v 2 = 1 .Set v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = c and v 2 h m 1 i q = b .Wenowhave v 2 t = c , v 2 u = b + 1 , v 2 v = c + b ,and v 2 j K j = cf + b + 1 g )]TJ/F71 11.9552 Tf 10.26 0 Td [(b )]TJ/F71 11.9552 Tf 10.26 0 Td [(c . Lemma19givesus U 0 M b + c + 1 , U M c + 1 ,and V s M b + 1 .WeuseLemma3to concludethefollowing. 1.If c < b ,wehave M c + 1 M b + 1 andthus V s M c + 1 .Alsosince U M c + 1 ,Theorem 23implies U ? M c + 1 .Hence dim M c + 1 g + 1 .AgainbyTheorem23 dim M b + 1 dim V s g ,and dim M b + c + 1 dim U 0 f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . So c + 1 g + 1 )]TJ/F71 11.9552 Tf 9.546 0 Td [(g + b + 1 g )]TJ/F68 11.9552 Tf 9.546 0 Td [( f )]TJ/F68 11.9552 Tf 9.546 0 Td [(1 + b + c + 1 f )]TJ/F68 11.9552 Tf 9.546 0 Td [(1 )]TJ/F68 11.9552 Tf 9.546 0 Td [(1 = v 2 j K j .NowbyLemma 3,setting j = 3 , s 1 = g + 1 , s 2 = g , s 3 = f )]TJ/F68 11.9552 Tf 10.23 0 Td [(1 , s 4 = dim ker L = 1 , t 1 = c + 1 , t 2 = b + 1 , and t 3 = b + c + 1 ,wehave e 0 = f , e c + 1 = 1 , e b + 1 = g )]TJ/F71 11.9552 Tf 12.177 0 Td [(f + 1 , e b + c + 1 = f )]TJ/F68 11.9552 Tf 10.384 0 Td [(2 ,and e i = 0 forallother i . 2.If c > b ,Byargumentssimilartothoseabovewecanshow dim M b + 1 g + 1 .Wealso have dim M c + 1 f ,and dim M b + c + 1 f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . 56

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So b + 1 g + 1 )]TJ/F71 11.9552 Tf 12.303 0 Td [(f + c + 1 f )]TJ/F68 11.9552 Tf 10.509 0 Td [( f )]TJ/F68 11.9552 Tf 10.51 0 Td [(1 + b + c + 1 f )]TJ/F68 11.9552 Tf 10.51 0 Td [(1 )]TJ/F68 11.9552 Tf 10.51 0 Td [(1 = v 2 j K j .Applying Lemma3asabove,weget e 0 = f , e c + 1 = 1 , e b + 1 = g )]TJ/F71 11.9552 Tf 12.187 0 Td [(f + 1 , e b + c + 1 = f )]TJ/F68 11.9552 Tf 10.395 0 Td [(2 ,and e i = 0 forallother i , 0 . 3.If b = c ,bysimilararguments,wecanshow e 0 = f , e c + 1 = g )]TJ/F71 11.9552 Tf 12.2 0 Td [(f + 2 , e 2 c + 1 = f )]TJ/F68 11.9552 Tf 10.407 0 Td [(2 ,and e i = 0 forallother i . 4.7.3 2 -ElementaryDivisorsof S and K when m IsOdd. 4.7.3.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A 2 with A .As m isodd,wehave v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = 1 and v 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q > 1 .Weset v 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = a , v 2 q )]TJ/F68 11.9552 Tf 10.852 0 Td [(1 = w and v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + 1 = d .So v 2 r = a + w , v 2 s = d , v 2 k = v 2 = a + d ,and v 2 j S j = a + w f + dg + a + d . ByLemma19,wehave C M a , U M a + w + d ,and V s M d .UsingLemma3wearriveat thefollowingconclusions. 1.Assumethat d < a ,then M d M a C .Nowsince V s and C aresubsetsof M d ,Theorem23givesus U 0? M d ,andthus dim M d g + 2 .Since C M a ,we have dim M a f + 1 .AgainbyTheorem23weget dim M a + w + d f = dim U . Now, d g + 2 )]TJ/F68 11.9552 Tf 10.208 0 Td [( f + 1 + a f + 1 )]TJ/F71 11.9552 Tf 12.002 0 Td [(f + a + w + d f = v 2 j S j .SobyLemma3,setting j = 3 , s 1 = g + 2 , s 2 = f + 1 , s 3 = f , s 4 = dim ker A , t 1 = d , t 2 = a ,and t 3 = a + w + d , wehave e 0 = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e d = g )]TJ/F71 11.9552 Tf 12.053 0 Td [(f + 1 , e a = 1 , e a + w + d = f ,and e i = 0 forallother i . 2.If a < d ,byargumentssimilartotheonesabove,wecanshow dim M a g + 2 . As U M a + w + d M d and V s M d ,Theorem23implies dim M d g + 1 ,and dim M a + w + d f . Now, a g + 2 )]TJ/F68 11.9552 Tf 10.445 0 Td [( g + 1 + d g + 1 )]TJ/F71 11.9552 Tf 12.237 0 Td [(f + a + w + d f = v 2 j S j .ApplyingLemma3as above,wehave e 0 = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e d = g )]TJ/F71 11.9552 Tf 12.053 0 Td [(f + 1 , e a = 1 , e a + w + d = f ,and e i = 0 forallother i . 3.If a = d ,bysimilarargumentswecanshowthat e 0 = f )]TJ/F68 11.9552 Tf 10.465 0 Td [(1 , e a = g + 2 )]TJ/F71 11.9552 Tf 12.258 0 Td [(f , e a + d + w = f , and e i = 0 forallother i . 57

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4.7.3.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L 2 with L .As m is odd,wehave v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = 1 and v 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q > 1 .Set v 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = a and v 2 s = v 2 q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + 1 = d . So v 2 t = a + 1 , v 2 u = d , v 2 v = 1 ,and v 2 j K j = a + 1 f + dg )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . ByLemma19,wehave C M a and U M a + d + 1 .AgainbyLemma19wehave V s M d . UsingLemma3wearriveatthefollowingconclusions. 1.Assume d < a ,then C M a M d .As V s M d ,Theorem23implies U 0 ? M d , andhence dim M d dim U 0? = g + 2 .Also dim M a dim C = f + 1 ,and dim M a + d + 1 dim U = f . Now, d g + 2 )]TJ/F68 11.9552 Tf 9.802 0 Td [( f + 1 + a f + 1 )]TJ/F71 11.9552 Tf 11.595 0 Td [(f + a + d + 1 f )]TJ/F68 11.9552 Tf 9.802 0 Td [(1 = dg + a + 1 f )]TJ/F71 11.9552 Tf 9.801 0 Td [(a )]TJ/F68 11.9552 Tf 9.802 0 Td [(1 = v 2 j K j . SobyLemma3,setting j = 3 , s 1 = g + 2 , s 2 = g + 1 , s 3 = f , s 4 = dim ker L = 1 , t 1 = a , t 2 = d ,and t 3 = a + d + 1 ,wehave e 0 = f )]TJ/F68 11.9552 Tf 10.298 0 Td [(1 , e a = 1 , e d = g + 1 )]TJ/F71 11.9552 Tf 12.091 0 Td [(f , e a + d + 1 = f )]TJ/F68 11.9552 Tf 10.298 0 Td [(1 ,and e i = 0 forall i . 2.If a < d ,byargumentssimilartothoseabove, dim M a g + 2 .As 1 2 M d , and V s M d ,byTheorem23,itfollowsthat dim M d g + 1 .Wealsohave dim M a + d + 1 f . Now, a g + 2 )]TJ/F68 11.9552 Tf 9.252 0 Td [( g + 1 + d g + 1 )]TJ/F71 11.9552 Tf 11.045 0 Td [(f + a + d + 1 f )]TJ/F68 11.9552 Tf 9.252 0 Td [(1 = dg + a + 1 f )]TJ/F71 11.9552 Tf 9.252 0 Td [(a )]TJ/F68 11.9552 Tf 9.252 0 Td [(1 = v 2 j K j .By applyingLemma3asabove,wehave e 0 = f )]TJ/F68 11.9552 Tf 10.363 0 Td [(1 , e a = 1 , e d = g + 1 )]TJ/F71 11.9552 Tf 12.157 0 Td [(f , e a + d + 1 = f )]TJ/F68 11.9552 Tf 10.363 0 Td [(1 , and e i = 0 forallother i . 3.If a = d ,byargumentssimilartothoseabovewemayshowthat e 0 = f )]TJ/F68 11.9552 Tf 8.493 0 Td [(1 , e a = g + 2 )]TJ/F71 11.9552 Tf 10.287 0 Td [(f , e 2 a + 1 = f )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 ,and e i = 0 forallother i . 4.7.4 ` -ElementaryDivisorsof S and K when m isOdd,and ` j q + 1 . 4.7.4.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A ` with A .In thiscase ` j q + 1 isanoddprimewith m odd.Sowehave v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = v ` r .Weset v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = v ` r = a ,and v ` s = v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + 1 = d .Then v ` k = a + d = v ` and v ` j S j = af + dg + a + d . ByLemma19,wehave U M a + d , V s M d ,and V r M a .WenowapplyTheorem23 andLemma3toconcludethefollowing. 58

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1.Assume a < d ,then V s M a M d .Since V r M a ,Theorem23implies dim M a dim U ? = g + 1 .As U M a + d M a ,Theorem23implies dim M a dim U ? g + 1 . Fromabove,wehave dim M a + d dim U = f . Now, a g + 2 )]TJ/F68 11.9552 Tf 10.491 0 Td [( g + 1 + d g + 1 )]TJ/F71 11.9552 Tf 12.284 0 Td [(f + a + d f = v ` j S j .ByLemma3,setting j = 3 , s 1 = g + 2 , s 2 = g + 1 , s 3 = f , s 4 = dim ker A = 1 , t 1 = a , t 2 = d ,and t 3 = a + d ,wehave e 0 = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e a = 1 , e d = g )]TJ/F71 11.9552 Tf 12.054 0 Td [(f + 1 , e a + d = f ,and e i = 0 forallother i . 2.If d < a ,byargumentssimilartothoseabove,wecanshowthat dim M d g + 1 .Since U M a + d M a and V r M a ,Theorem23implies dim M a dim C f + 1 .From above,wehave dim M a + d dim U = f . Now, b g + 2 )]TJ/F68 11.9552 Tf 10.177 0 Td [( f + 1 + d f + 1 )]TJ/F71 11.9552 Tf 11.97 0 Td [(f + a + d f = v ` j S j .ByapplyingLemma3,weget e 0 = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e a = 1 , e d = g )]TJ/F71 11.9552 Tf 12.054 0 Td [(f + 1 , e a + d = f ,and e i = 0 forallother i . 3.If a = d ,byargumentssimilartothoseabove,weget e 0 = f )]TJ/F68 11.9552 Tf 9.736 0 Td [(1 , e a = g )]TJ/F71 11.9552 Tf 11.528 0 Td [(f + 2 , e a + d = f , and e i = 0 forallother i . 4.7.4.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L ` with L .Inthis case ` j q + 1 isanoddprimewith m odd,wehave v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q > 0 , v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = 0 ,and v ` h m 1 i q = 0 .Setting v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q = av 2 s = v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + 1 = d ,wehave v ` t = a , v ` u = c , v ` v = 0 ,and v ` j K j = af + dg .As L )]TJ/F71 11.9552 Tf 20.606 0 Td [(A mod2 a + d ,wehave M i L = M i A forall i a + c .Sowehave U M a + d , V s M d ,and V r M a UsingthisfactandLemma3weconcludethefollowing. 1.If a < d ,then M a M d V s .Since M a V r aswell,byTheorem23wehave dim M a g + 2 , dim M d g + 1 ,and dim M a + d f . Now a g + 2 )]TJ/F68 11.9552 Tf 10.347 0 Td [( g + 1 + d g + 1 )]TJ/F71 11.9552 Tf 12.141 0 Td [(f + a + b f )]TJ/F68 11.9552 Tf 10.348 0 Td [(1 = v ` j K j .SobyLemma3,setting j = 3 , s 1 = g + 2 , s 2 = g + 1 , s 3 = f , s 4 = dim ker L = 1 , t 1 = a , t 2 = d ,and t 3 = a + d , wehave e 0 = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e a = 1 , e d = g )]TJ/F71 11.9552 Tf 12.053 0 Td [(f + 1 ,and e a + d = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e i = 0 forallother i . 2.If a > d ,wehave M d M a .Sobysimilararguments dim M d g + 2 .Andbytheabove wehave dim M a g + 1 ,and dim M a + d f . 59

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Now a g + 2 )]TJ/F68 11.9552 Tf 10.438 0 Td [( f + 1 + d f + 1 )]TJ/F71 11.9552 Tf 12.231 0 Td [(f + a + d f )]TJ/F68 11.9552 Tf 10.438 0 Td [(1 = v ` j K j .ByLemma3,wehave e 0 = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e a = 1 , e d = g )]TJ/F71 11.9552 Tf 12.054 0 Td [(f + 1 ,and e a + d = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e i = 0 forallother i . 3.If a = d ,byargumentssimilartothoseabove,wehave e 0 = f )]TJ/F68 11.9552 Tf 10.883 0 Td [(1 , e a = g )]TJ/F71 11.9552 Tf 12.676 0 Td [(f + 2 , e 2 a = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e i = 0 forallother i . 4.7.5 ` -ElementaryDivisorsof K when m IsEvenand ` j q + 1 . Inthiscasewehave q )]TJ/F68 11.9552 Tf 20.005 0 Td [(1mod ` andthus r s )]TJ/F68 11.9552 Tf 20.005 0 Td [(2mod ` andthus ` j S j . Howeverweseethat ` j t and ` j u andthus ` jj K j .Inthissectionwecomputethe ` -elementarydivisorsof K . 4.7.5.0.1Elementarydivisorsof K . .Weidentify M i L with M i and L ` with L .Set v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = c and v ` h m 1 i q = b .Since m isodd,wehave v ` s = v ` q m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + 1 = 0 .So v ` t = c , v ` u = b , v ` v = c + b , v ` = 0 ,and v ` j K j = cf + bg )]TJ/F68 11.9552 Tf 10.26 0 Td [( c + b . Lemma19givesus V r M c , V s M b and U 0 M b + c .WenowuseLemma3to concludethefollowing. 1.If c < b ,wehave M c M b andthus V s M c .Alsosince V r M c ,Theorem23implies U ? M c .Hence dim M c g + 1 .AgainbyTheorem23 dim M b dim V s g ,and dim M b + c dim U 0 f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . So c g + 1 )]TJ/F71 11.9552 Tf 9.722 0 Td [(g + b g )]TJ/F68 11.9552 Tf 9.721 0 Td [( f )]TJ/F68 11.9552 Tf 9.722 0 Td [(1 + b + c f )]TJ/F68 11.9552 Tf 9.721 0 Td [(1 )]TJ/F68 11.9552 Tf 9.722 0 Td [(1 = v 2 j K j .NowbyLemma3,setting j = 3 , s 1 = g + 1 , s 2 = g , s 3 = f )]TJ/F68 11.9552 Tf 10.4 0 Td [(1 , s 4 = dim ker L = 1 , t 1 = c , t 2 = b ,and t 3 = b + c , wehave e 0 = f , e c = 1 , e b = g )]TJ/F71 11.9552 Tf 12.053 0 Td [(f + 1 , e b + c = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 ,and e i = 0 forallother i . 2.If c > b ,Byargumentssimilartothoseabovewecanshow dim M b g + 1 .Wealso have dim M c f ,and dim M b + c f )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . So b g + 1 )]TJ/F71 11.9552 Tf 12.174 0 Td [(f + c f )]TJ/F68 11.9552 Tf 10.382 0 Td [( f )]TJ/F68 11.9552 Tf 10.381 0 Td [(1 + b + c f )]TJ/F68 11.9552 Tf 10.381 0 Td [(1 )]TJ/F68 11.9552 Tf 10.381 0 Td [(1 = v 2 j K j .ApplyingLemma3as above,weget e 0 = f , e c = 1 , e b = g )]TJ/F71 11.9552 Tf 12.054 0 Td [(f + 1 , e b + c = f )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 ,and e i = 0 forallother i , 0 . 3.If b = c ,bysimilararguments,wecanshow e 0 = f , e c = g )]TJ/F71 11.9552 Tf 11.338 0 Td [(f + 2 , e 2 c = f )]TJ/F68 11.9552 Tf 9.545 0 Td [(2 ,and e i = 0 forallother i . 60

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4.8 ` -ElementaryDivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.572 -1.793 Td [(ue q ; m ,and ` j q + 1 . Giventhegraph )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(ue q ; m andaprime ` ,table4-1showsthat A ` equivalently L ` is nilpotentifandonlyif ` j q + 1 .Inthissectionwecomputethe ` -elementarydivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.572 -1.793 Td [(ue q ; m and ` j q + 1 . Weset h = 1 2 ,and z = m inLemma12andLemma11togetparametersforthisgraph. Thus )]TJ/F71 7.9701 Tf 6.647 -1.794 Td [(ue q ; m isanSRGwithparameters v = h m 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 , k = q 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 , = q 2 )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 + q 4 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(5 + 1 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q 2 ,and = h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 . Theadjacencymatrix A haseigenvalues k ; r ; s = k ; q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F68 11.9552 Tf 11.075 0 Td [(1 ; )]TJ/F68 11.9552 Tf 7.603 0 Td [( + q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 ,with multiplicities ; f ; g = 0 B B B B B B B B @ 1 ; q 2 h m 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 q + 1 ; q 3 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 1 C C C C C C C C A .So L haseigenvalues ; t ; u = 0 ; h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 + q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; h m 1 i q 2 + q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 withmultiplicities ; f ; g . 4.8.1SubmoduleStructure Wenowrecallfrom x 4.3thedenitionsof C , C 0 U , U 0 , V r and V s inthecontextofthe graph )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(ue .Inthiscase G V = U m ; q 2 .ByCorollary 2 : 10 ,Corollary 4 : 5 ,andLemma4.6of [27],wehavethefollowingresult. Theorem24. When ` j q + 1 ,themodule U 0? hasasubmodule M containing V s suchthat dim M = V s = 1 .Therelativepositionsof M , U ? , C , V s , V r , U , U 0 ,and h 1 i inthe F ` U m ; q 2 submodulestructureof U 0 ? areasintheHassediagramsgiveningure4-5.Here : = dim V r = U = q 2 m )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q + 1 , y : = dim V s = U 0 = q 2 m )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F71 11.9552 Tf 10.261 0 Td [(q q + 1 2 , x : = dim U 0 = q 2 m )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 q 2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and dim M = V s = dim U = U 0 = dim C = V r = 1 . 4.8.2 ` -ElementaryDivisorsof S and K when ` j q + 1 and ` m . 4.8.2.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A ` with A .Inthis case, ` j q + 1 and ` m .Set v ` q 2 )]TJ/F68 11.9552 Tf 10.763 0 Td [(1 = w , v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 = a ,and v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 = d .Then v ` r = w + a , v ` s = d , v ` k = v ` = a + d ,and v ` j S j = w + a f + dg + a + d . Now s isaneigenvalueofvaluation d and k isaneigenvalueofvaluation a + d .So Lemma4implies V s M d and h 1 i M a + d M d .SobyTheorem24,wehave M d V s h 1 i . Lemma19implies C M a , U M w + a + d ,and V r M w + a . 61

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` m U 0 ? U ? C M V r V s U h 1 i U 0 ` j m U 0? U ? C M V r V s U U 0 h 1 i Figure4-5.Submodulestructure. Foranypositiveinteger n ,wehave q 2 n + 1 + 1 = q + 1 h 2 n + 1 1 i )]TJ/F71 7.9701 Tf 5.069 0 Td [(q .Thereforeif ` j q + 1 ,the followingaretrue. 1 v ` q 2 n + 1 + 1 = v ` q + 1 ifandonlyif ` j 2 n + 1 . 2 v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 = 0 ifandonlyif ` j m )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . 3 v ` q 2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 = v ` q + 1 ifandonlyif ` , 2 . Subcase1:When ` m )]TJ/F68 11.9552 Tf 11.585 0 Td [(1 . Inthiscase, a = 0 ,since v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 = 0 .Wehave v ` j S j = wf + dg + d .WeapplyLemma3andTheorem24toarriveatthefollowingresults. 1.Assume w < d ,thenwehaveas M d V s h 1 i ,and V r M d M w V r .Soby Theorem24, U ? = V r + V s M w .Wesawthat M w + d U .AgainbyTheorem24,we have dim M w dim U ? = f + g + 1 )]TJ/F68 11.9552 Tf 10.724 0 Td [(dim U = f + g )]TJ/F71 11.9552 Tf 11.322 0 Td [(x , dim M d g + 1 ,and dim M w + d dim U = x + 1 . Now a f + g )]TJ/F71 11.9552 Tf 10.197 0 Td [(x )]TJ/F68 11.9552 Tf 9.6 0 Td [( g + 1 + d g + 1 )]TJ/F68 11.9552 Tf 9.6 0 Td [( x + 1 + w + d x + 1 = wf + dg + d = v ` j S j .So byLemma3,setting j = 3 , s 1 = f + g )]TJ/F71 11.9552 Tf 11.012 0 Td [(x , s 2 = g + 1 , s 3 = x + 1 , s 4 = dim ker A = 0 , 62

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t 1 = w , t 2 = d ,and t 3 = w + d ,wehave e 0 = x + 1 , e w = f )]TJ/F71 11.9552 Tf 10.438 0 Td [(x )]TJ/F68 11.9552 Tf 9.841 0 Td [(1 , e d = g )]TJ/F71 11.9552 Tf 10.439 0 Td [(x , e w + d = x + 1 , and e i = 0 forallother i . 2.If w > d ,byargumentssimilartothoseabovewecanshowthat M d U ? , M w V r , and M w + d U .ApplyingLemma3asabove,wecanconcludethat e 0 = x + 1 , e w = f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e d = g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x , e w + d = x + 1 ,and e i = 0 forallother i . 3.If w = d ,againbyargumentssimilartothoseabove,wecanshowthat e 0 = x + 1 , e w = f + g )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e w + d = x + 1 ,and e i = 0 forallother i . Subcase2:When ` j m )]TJ/F68 11.9552 Tf 10.747 0 Td [(1 . Inthiscase, a , 0 ,but ` 2 m )]TJ/F68 11.9552 Tf 10.746 0 Td [(3 .So d = v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 = v ` q + 1 v ` q 2 )]TJ/F68 11.9552 Tf 10.72 0 Td [(1 = w ,withtheequalityholdingifandonlyif ` , 2 .Sowehaveeither a d < w + a < w + a + d ,or d < a < w + a < w + a + d . 1.If a < d < w + a < w + a + d ,wehave M d M a .SobyTheorem24 M a M d C + V s = U 0 ? .Since M w + a V r and d < w + a ,Theorem24implies M d V s + V r = U ? .Wealso have M w + a V r and M w + a + d U .Thuswehave dim M a dim U 0? = f + g + 1 )]TJ/F71 11.9552 Tf 11.053 0 Td [(x , dim M w dim U ? = f + g )]TJ/F71 11.9552 Tf 10.857 0 Td [(x , dim M w + a f ,and dim M w + a + d x + 1 . Now, a f + g + 1 )]TJ/F71 11.9552 Tf 10.857 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [( f + g )]TJ/F71 11.9552 Tf 10.857 0 Td [(x + d f + g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F71 11.9552 Tf 12.053 0 Td [(f + w + d f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 + w + a + d x + 1 = w + a f + dg + d + a = v ` j S j : ThusbyLemma3,setting j = 4 , s 1 = f + g + 1 )]TJ/F71 11.9552 Tf 11.064 0 Td [(x , s 2 = f + g )]TJ/F71 11.9552 Tf 11.065 0 Td [(x , s 3 = f , s 4 = x + 1 , s 5 = dim ker A = 0 , t 1 = a , t 2 = d , t 3 = w + a ,and t 4 = w + a + d ,weconcludethat e 0 = x , e a = 1 , e d = g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x , e w + a = f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e w + a + d = x + 1 ,and e i = 0 forallother i . 2.If d < a < w + a < w + a + d ,byargumentssimilartothoseabove,wecanshow M d U 0 ? , M a C M w + d V r ,and M w + a + d U .NowbyapplyingLemma3like above,wehave e 0 = x , e a = 1 , e d = g )]TJ/F71 11.9552 Tf 10.884 0 Td [(x , e w + a = f )]TJ/F71 11.9552 Tf 10.884 0 Td [(x )]TJ/F68 11.9552 Tf 10.286 0 Td [(1 , e w + a + d = x + 1 ,and e i = 0 for allother i . 63

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3.If d = a < w + a < w + a + d ,bysimilararguments, M a U 0 ? , M w + a V r , and M w + a + d U .NowbyapplyingLemma3likeabove,wecanshowthat e 0 = x , e a = e d = g + 1 )]TJ/F71 11.9552 Tf 10.858 0 Td [(x , e w + a = f )]TJ/F71 11.9552 Tf 10.857 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e w + a + d = x + 1 ,and e i = 0 forallother i . 4.8.2.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L ` with L . Weset v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 = a , v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 = d ,and v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = c .Then v ` k = v ` = a + d , v ` v = v ` h m 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = c .So v ` t = a + c , v ` u = d ,and v ` j K j = a + c f + dg )]TJ/F71 11.9552 Tf 10.26 0 Td [(c . As L 1 = 0 ,wehave h 1 i M i forall i .Since s isaneigenvalueofvaluation d ,Lemma4 implies M d V r .SobyTheorem24,weseethat M d V s h 1 i .Since t isaneigenvalueof valuation a + c ,Lemma4implies V r M a + c .Lemma19givesus C M a and U 0 M a + c + d . ThusbyTheorem24, U 0 h 1 i = U M a + c + d . Subcase1:When ` m )]TJ/F68 11.9552 Tf 8.831 0 Td [(1 . Inthiscase, a = 0 ,as v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 = 0 .Thus v ` j K j = cf + dg )]TJ/F71 11.9552 Tf 8.831 0 Td [(c . WeapplyLemma4,andTheorem24toconcludethefollowing. 1.If c < d < d + c ,Fromtheinformationwegatheredabove,wehave M c V r and M c M d V s h 1 i .ApplyingTheorem24givesus M c U ? andhence dim M c f + g )]TJ/F71 11.9552 Tf 10.454 0 Td [(x . WealsohavebyTheorem24, dim M d g + 1 and dim M d + c x + 1 . Nowwehave, c f + g )]TJ/F71 11.9552 Tf 8.994 0 Td [(x )]TJ/F68 11.9552 Tf 8.396 0 Td [( g + 1 + d g + 1 )]TJ/F68 11.9552 Tf 8.396 0 Td [( x + 1 + c + d x + 1 )]TJ/F68 11.9552 Tf 8.396 0 Td [(1 = cf + dg )]TJ/F71 11.9552 Tf 8.396 0 Td [(c = v ` j K j . SobyLemma3,setting j = 3 , s 1 = f + g )]TJ/F71 11.9552 Tf 10.215 0 Td [(x , s 2 = g + 1 , s 3 = x + 1 , s 4 = dim ker L = 1 , t 1 = c , t 2 = d ,and t 3 = c + d ,wehave e 0 = x + 1 , e c = f )]TJ/F71 11.9552 Tf 10.866 0 Td [(x )]TJ/F68 11.9552 Tf 10.268 0 Td [(1 , e d = g )]TJ/F71 11.9552 Tf 10.866 0 Td [(x , e c + d = x ,and e i = 0 forallother i . 2.If d < c < d + c ,Byargumentssimilartothoseabove,wecanshow M d U ? , and dim M d f + g )]TJ/F71 11.9552 Tf 11.761 0 Td [(x ; M c V r ,and dim M c f + 1 ;and M d + c U ,and dim M d + c x + 1 .ByapplyingLemma3asabove,wecanshowthat e 0 = x + 1 , e c = f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 , e d = g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x , e c + d = x ,and e i = 0 forallothernon-zero i . 3.If c = d < d + c ,byargumentssimilartothoseabovewecanshow e 0 = x + 1 , e c = f + g )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 x )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 , e c + d = x ,and e i = 0 forallother i . 64

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Subcase2:When ` j m )]TJ/F68 11.9552 Tf 10.904 0 Td [(1 . As q )]TJ/F68 11.9552 Tf 20.137 0 Td [(1mod ` and ` j m )]TJ/F68 11.9552 Tf 10.904 0 Td [(1 ,bytheobservationsat thebeginningofthesubsection,wehave c = d .WeapplyLemma3andTheorem24to concludethefollowing. 1.Assumethat a < c = d < a + d < a + c + d .As C M a ,and V s M c M d ,by Theorem24 M d U 0 ? ,andthus dim M d f + g + 1 )]TJ/F71 11.9552 Tf 10.904 0 Td [(x .AlsobyTheorem24,since V r M a + d M c , M c U ? andthus dim M c f + g )]TJ/F71 11.9552 Tf 11.134 0 Td [(x .Wealsohave U M a + c + d ; V r M a + d ,andthus dim M d + a f ,and dim M a + c + d x + 1 . Wehave a f + g + 1 )]TJ/F71 11.9552 Tf 8.643 0 Td [(x )]TJ/F68 11.9552 Tf 8.046 0 Td [( f + g )]TJ/F71 11.9552 Tf 8.643 0 Td [(x + c f + g )]TJ/F71 11.9552 Tf 8.644 0 Td [(x )]TJ/F68 11.9552 Tf 8.045 0 Td [( f + a + d f )]TJ/F68 11.9552 Tf 8.046 0 Td [( x + 1 + a + c + d x + 1 )]TJ/F68 11.9552 Tf 8.046 0 Td [(1 = a + d f + cg )]TJ/F68 11.9552 Tf 10.742 0 Td [( a + d = v ` j K j .SobyLemma3,setting j = 4 , s 1 = f + g + 1 )]TJ/F71 11.9552 Tf 11.339 0 Td [(x , s 2 = f + g )]TJ/F71 11.9552 Tf 11.173 0 Td [(x , s 3 = f , s 4 = x + 1 s 5 = dim ker L = 1 , t 1 = a , t 2 = c , t 3 = a + d ,and t 4 = a + c + d ,wehave e 0 = x , e a = 1 , e c = g )]TJ/F71 11.9552 Tf 10.923 0 Td [(x , e a + d = f )]TJ/F71 11.9552 Tf 10.924 0 Td [(x )]TJ/F68 11.9552 Tf 10.325 0 Td [(1 , e a + c + d = x ,and e i = 0 forallothernon-zero i . 2.Assumethat c = d < a < a + d < b + c + d .Thenbyargumentssimilartothoseabove, M c C + V s = U 0 ? , M a C , M a + d V r ,and M a + c + d U .NowapplyingLemma3as above,wehave e 0 = x , e a = 1 , e c = g )]TJ/F71 11.9552 Tf 10.996 0 Td [(x , e a + d = f )]TJ/F71 11.9552 Tf 10.995 0 Td [(x )]TJ/F68 11.9552 Tf 10.398 0 Td [(1 , e a + c + d = x ,and e i = 0 forall othernon-zero i . 3.If c = d = a < a + d < a + c + d ,thenbyargumentssimilartothoseabove,wecanshow e 0 = x , e c = g + 1 )]TJ/F71 11.9552 Tf 10.858 0 Td [(x , e a + d = f )]TJ/F71 11.9552 Tf 10.857 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e a + c + d = x ,and e i = 0 forallother i . 4.8.3 ` -ElementaryDivisorsof S and K when ` j q + 1 ,and ` j m . 4.8.3.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A ` with A .Inthis case ` m )]TJ/F68 11.9552 Tf 10.086 0 Td [(1 andthus ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 and v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F68 11.9552 Tf 10.085 0 Td [(1 = v ` q 2 )]TJ/F68 11.9552 Tf 10.086 0 Td [(1 .As ` 2 m )]TJ/F68 11.9552 Tf 10.085 0 Td [(3 and ` 2 m )]TJ/F68 11.9552 Tf 10.086 0 Td [(1 , wehave v ` s = v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 = v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = v ` q + 1 v ` q 2 )]TJ/F68 11.9552 Tf 10.175 0 Td [(1 .Set w = v ` q 2 )]TJ/F68 11.9552 Tf 10.174 0 Td [(1 = v ` r , v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 = v ` s = d . Wehave v ` j S j = wf + dg + d ,and v ` k = v ` = d Observethat d w < d + w .As r isaneigenvalueofvaluation w d and s ; k are eigenvaluesofvaluation d ,byLemma4andTheorem24, 65

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M d V r + V s = U ? ,and M w V r .ByLemma19, U M w + d .ThusbyTheorem24, dim M d f + g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x , dim M w f and dim M w + d x + 1 . Wehave d f + g )]TJ/F71 11.9552 Tf 10.053 0 Td [(x )]TJ/F71 11.9552 Tf 11.248 0 Td [(f + w f )]TJ/F68 11.9552 Tf 9.455 0 Td [( x + 1 + d + w x + 1 = v ` j S j .SobyLemma3,setting j = 3 , s 1 = f + g )]TJ/F71 11.9552 Tf 10.549 0 Td [(x , s 2 = f , s 3 = x + 1 , s 4 = dim ker A = 0 , t 1 = d , t 2 = w ,and t 3 = w + d ,we have e 0 = x + 1 , e d = g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x , e w = f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 , e w + d = x + 1 ,and e i = 0 forallother i . 4.8.3.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L ` with L .Since ` m )]TJ/F68 11.9552 Tf 10.212 0 Td [(1 ,wehave ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 andthus v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F68 11.9552 Tf 10.212 0 Td [(1 = v ` q 2 )]TJ/F68 11.9552 Tf 10.211 0 Td [(1 .As ` 2 m )]TJ/F68 11.9552 Tf 10.211 0 Td [(3 and ` 2 m )]TJ/F68 11.9552 Tf 10.212 0 Td [(1 , wehave v ` s = v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 = v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = v ` q + 1 v ` q 2 )]TJ/F68 11.9552 Tf 10.264 0 Td [(1 .Set v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = d and v ` h m 1 i q 2 = b . Wehave v ` k = v ` = d , v ` v = b + d , v ` t = d , v ` u = d + b ,and v ` j K j = df + d + b g )]TJ/F68 11.9552 Tf 10.26 0 Td [( b + d . As L 1 = 0 ,wehave h 1 i M i forall i .Since t isaneigenvalueofvaluation d , and u isaneigenvalueofvaluation b + d ,byLemma4andTheorem24,weseethat M d V r + V s = U ? and M d + b V s . ByLemma19,wehave U M 2 d and U 0 M b + 2 d . WeapplyLemma3andTheorem24wearriveatthefollowingconclusions. 1.If b < d , 2 d > b + d ,wehave M b + d M 2 d + V s U + V s .ThusbyTheorem24,wesee that M b + d M andthus dim M b + d g + 1 .Since M d U ? , M 2 d U and M b + 2 d U 0 , Theorem24implies dim M d f + g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x dim M 2 d x + 1 ,and dim M b + 2 d x . Now, d f + g )]TJ/F71 11.9552 Tf 9.875 0 Td [(x )]TJ/F68 11.9552 Tf 9.277 0 Td [( g + 1 + b + d g + 1 )]TJ/F68 11.9552 Tf 9.277 0 Td [( x + 1 + 2 d x + 1 )]TJ/F71 11.9552 Tf 9.875 0 Td [(x + b + 2 d x )]TJ/F68 11.9552 Tf 9.278 0 Td [(1 = v ` j K j . SobyLemma3,setting j = 4 , s 1 = f + g )]TJ/F71 11.9552 Tf 11.822 0 Td [(x , s 2 = g + 1 , s 3 = x + 1 , s 4 = x , s 5 = dim ker L = 1 , t 1 = d , t 2 = b + d , t 3 = 2 d ,and t 4 = b + 2 d ,wehave e 0 = x + 1 , e d = f )]TJ/F71 11.9552 Tf 10.858 0 Td [(x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e b + d = g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x , e 2 d = 1 , e b + 2 d = x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e i = 0 forallother i . 2.If b > d ,bysimilararguments, M 2 d M , M b + d V s , M d U ? , M 2 d U and M b + 2 d U 0 .ApplyingLemma3likeintheabovecase,wehave e 0 = x + 1 , e d = f )]TJ/F71 11.9552 Tf 9.403 0 Td [(x )]TJ/F68 11.9552 Tf 8.805 0 Td [(1 , e b + d = g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x , e 2 d = 1 , e b + 2 d = x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e i = 0 forallother i . 66

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3.If b = d ,byargumentssimilartothoseabove,wehave e 0 = x + 1 , e b = f )]TJ/F71 11.9552 Tf 11.497 0 Td [(x )]TJ/F68 11.9552 Tf 10.9 0 Td [(1 , e b + d = g )]TJ/F71 11.9552 Tf 10.858 0 Td [(x + 1 , e d + 2 b = x )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e i = 0 forallother i . 4.9 ` -ElementaryDivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.572 -1.793 Td [(uo q ; m ,and ` j q + 1 . Giventhegraph )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m andaprime ` ,table4-1showsthat A ` equivalently L ` is nilpotentifandonlyif ` j q + 1 .Inthissectionwecomputethe ` -elementarydivisorsof S and K when )]TJ/F68 11.9552 Tf 6.647 0 Td [( V =)]TJ/F71 7.9701 Tf 17.572 -1.794 Td [(uo q ; m and ` j q + 1 . Fromnowoninthissection,wedenote )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m by )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo ,and ` isaprimethatmeets thedescriptioninthepreviousparagraph.Weset h = 3 2 ,and z = m inLemma12and Lemma11togetparametersforthisgraph.Thus )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo q ; m isanSRGwithparameters v = h m 1 i q 2 q 2 m + 1 + 1 , k = q 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 , = q 2 )]TJ/F68 11.9552 Tf 11.091 0 Td [(1 + q 4 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 + 1 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 1 i q 2 ,and = h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 . Theadjacencymatrix A haseigenvalues k ; r ; s = k ; q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F68 11.9552 Tf 11.146 0 Td [(1 ; )]TJ/F68 11.9552 Tf 7.604 0 Td [( + q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 with multiplicities ; f ; g = 0 B B B B B B B B @ 1 ; q 3 h m 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 q + 1 ; q 2 h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 1 C C C C C C C C A .SotheLaplacian L has eigenvalues ; t ; u = ; k )]TJ/F71 11.9552 Tf 10.307 0 Td [(r ; k )]TJ/F71 11.9552 Tf 10.905 0 Td [(s = 0 ; h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 + q 2 m + 1 ; h m 1 i q 2 + q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 withmultiplicities ; f ; g . 4.9.1SubmoduleStructure Wenowrecallfrom x 4.3thedenitionsof C , C 0 U , U 0 , V r and V s inthecontextofthe graph )]TJ/F71 7.9701 Tf 6.647 -1.793 Td [(uo .Inthiscase G V = U m + 1 ; q 2 .ByCorollary 2 : 10 ,Corollary 5 : 6 ,andProposition 5 : 14 of[27],wehavethefollowingresult. Theorem25. If ` j q + 1 ,thefollowingaretrue. 1.If ` m ,then C V r U = h 1 i V s V s = U 0 . 2.If ` j m ,then C V r U V s = U 0 h 1 i . 3.Wehave dim C = f + 1 and dim U = g + 1 . 67

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4.9.2ElementaryDivisorsof S and K ,when ` m ,and ` j q + 1 4.9.2.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A ` with A .Weset v ` q 2 )]TJ/F68 11.9552 Tf 10.742 0 Td [(1 = w , v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 = a and v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = d .Sowehave v ` r = w + a , v ` s = d , v ` k = v ` = a + d ,and v ` j S j = w + a f + dg + a + d . Wehave a < w + a < w + a + d .As r isaneigenvalueofvaluation w + a ,byLemma 4,wehave M a + w V r .ByLemma19, M a C ,and M w + a + d U .ThusbyTheorem25, dim M a f + 1 , dim M w + a f ,and dim M w + a + d g + 1 . Now, a f + 1 )]TJ/F71 11.9552 Tf 11.335 0 Td [(f + w + a f )]TJ/F68 11.9552 Tf 9.541 0 Td [( g + 1 + w + a + d g + 1 = w + a f + dg + a + w = v ` j S j . SobyLemma3,setting j = 3 , s 1 = f + 1 , s 2 = f , s 3 = g + 1 , s 4 = dim ker A = 0 , t 1 = a , t 2 = w + a ,and t 3 = w + a + d ,wehave e 0 = g , e a = 1 , e w + a = f )]TJ/F71 11.9552 Tf 10.503 0 Td [(g )]TJ/F68 11.9552 Tf 10.503 0 Td [(1 , e w + a + d = g + 1 ,and e i = 0 forallother i . 4.9.2.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L ` with L .Weset v ` h m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 1 i q 2 = a , v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = c ,and v ` s = v ` q 2 m + 1 + 1 = d Sowehave v ` t = a + d , v ` u = c , v ` v = d ,and v ` j K j = a + d f + cg )]TJ/F71 11.9552 Tf 10.261 0 Td [(d . ByLemma19wehave C M a and U M a + d + c .As t isaneigenvalueofvaluation a + d byLemma4,wehave M a + d V t = V r .ByTheorem25,wehave dim M a f + 1 , dim M a + d f ,and dim M a + c + d g + 1 . Now a f + 1 )]TJ/F71 11.9552 Tf 12.277 0 Td [(f + a + d f )]TJ/F68 11.9552 Tf 10.484 0 Td [( g + 1 + a + c + d g + 1 )]TJ/F68 11.9552 Tf 10.484 0 Td [(1 = v ` j K j .SobyLemma 3,setting j = 3 , s 1 = f + 1 , s 2 = f , s 3 = g + 1 , s 4 = dim ker L = 1 , t 1 = a , t 2 = a + b ,and t 3 = a + b + c ,wehave e 0 = g , e a = 1 , e d + a = f )]TJ/F71 11.9552 Tf 10.26 0 Td [(g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e d + a + c = g ,and e i = 0 forallother i . 4.9.3ElementaryDivisorsof S and K ,when ` j m ,and ` j q + 1 . 4.9.3.0.1Elementarydivisorsof S . .Weidentify M i A with M i and A ` with A .As ` j m and q )]TJ/F68 11.9552 Tf 19.006 0 Td [(1mod ` ,wehave v ` q 2 )]TJ/F68 11.9552 Tf 10.451 0 Td [(1 = v ` r ,and v ` s = v ` k = .Set v ` q 2 )]TJ/F68 11.9552 Tf 10.451 0 Td [(1 = w and v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 = d .Wehave v ` j S j = wf + dg + d . As r isaneigenvalueofvaluation w ,Lemma4implies V r M w .ByTheorem19,we have M w + d U .ByTheorem25,wehave dim M w f ,and dim M w + d g + 1 . 68

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Now, w f )]TJ/F68 11.9552 Tf 10.711 0 Td [( g + 1 + w + d g + 1 = v ` j S j .SobyLemma3,setting j = 2 , s 1 = f , s 2 = g + 1 , s 3 = dim ker A = 0 , t 1 = w ,and t 2 = w + d ,wehave e 0 = g + 1 , e w = f )]TJ/F71 11.9552 Tf 10.472 0 Td [(g )]TJ/F68 11.9552 Tf 10.473 0 Td [(1 , e w + d = g + 1 ,and e i = 0 forallother i . 4.9.3.0.2Elementarydivisorsof K . .Weidentify M i L with M i and L ` with L .As ` j m ,wehave v ` q 2 m + 1 + 1 = v ` q 2 m )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 + 1 .Weset v ` h m 1 i q 2 = b and v ` q 2 m + 1 + 1 = d .We have v ` t = d , v ` u = b + d , v ` v = b + d ,and v ` j K j = df + b + d g )]TJ/F68 11.9552 Tf 10.26 0 Td [( b + d . As t isaneigenvalueofvaluation d ,wehave M d V r .Lemma19givesus M b + 2 d U 0 and U M 2 d .ByTheorem25,wehave dim M d f , dim M 2 d g + 1 ,and dim M b + 2 d g . Now, d f )]TJ/F68 11.9552 Tf 10.397 0 Td [( g + 1 + d g + 1 )]TJ/F71 11.9552 Tf 10.397 0 Td [(g + b + 2 d g )]TJ/F68 11.9552 Tf 10.397 0 Td [(1 = v ` j K j .SobyLemma3,setting j = 3 , s 1 = f , s 2 = g + 1 , s 3 = g , s 4 = dim ker L = 1 , t 1 = d , t 2 = 2 d ,and t 3 = b + 2 d ,wehave e 0 = g + 1 , e d = f )]TJ/F71 11.9552 Tf 10.26 0 Td [(g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 , e 2 d = 1 , e b + 2 d = g )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,and e i = 0 forallother i . 69

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CHAPTER5 PROOFSOFTHEOREM15ANDTHEOREM16. Theproofsinthischapterhavebeenreproducedfrommypaper“Criticalgroupsofvan Lint-SchrijverCyclotomicStronglyRegularGraphs”cf.[24]. 5.1SomePropertiesof G p ;`; t . Insection2of[31],theauthorsshowthat G p ;`; t isastronglyregulargraph. Theorem vanLint-Schrijver . Thegraph G p ;`; t isastronglyregulargraphwithparameters q ; q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ` ; q )]TJ/F68 11.9552 Tf 10.26 0 Td [(3 ` + 1 + )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 t + 1 ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ` )]TJ/F68 11.9552 Tf 10.261 0 Td [(2 p q ` 2 ; q )]TJ/F70 11.9552 Tf 10.26 0 Td [(` + 1 + )]TJ/F68 11.9552 Tf 7.603 0 Td [(1 t ` )]TJ/F68 11.9552 Tf 10.261 0 Td [(2 p q ` 2 ! ; where q = p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t .Theeigenvaluesoftheadjacencymatrix A of G p ;`; t are k = q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ` , r 1 , r ,withmultiplicities 1 , k ,and q )]TJ/F71 11.9552 Tf 11.044 0 Td [(k )]TJ/F68 11.9552 Tf 11.044 0 Td [(1 respectively.Here r = )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 + )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 t p q ` and r 1 = r + )]TJ/F68 11.9552 Tf 7.603 0 Td [(1 t + 1 p q . Wenowgiveabriefsketchoftheproofoftheabovegivenin x 2of[31].Werecallfrom x 3.1that K = F q and S istheuniquesubgroupof K ofsize k .Given a 2 S , b 2 K , n 2 Z , dene T a ; b ; n : K ! K by T a ; b ; n x : = ax p n + b .Let G bethegroupoftransformations T a ; b ; n . Theactionof G on K isshowntobeapermutationactionofrank 3 .Itisalsoshownthatthe orbitsofthenaturalactionof G on K K are f x ; x j x 2 K g , : = f x ; y j x ; y 2 K and x )]TJ/F71 11.9552 Tf 8.954 0 Td [(y 2 S g and : = f x ; y j x ; y 2 K and x )]TJ/F71 11.9552 Tf 10.353 0 Td [(y < S [f 0 gg .Thegraph G p ;`; t isthegraphwithvertexset K andedgeset .Standardresultsonrank 3 permutationgroupsofevenordershowthat G p ;`; t isastronglyregulargraph. Followingnotationin x 3.1,wehave A [ x ] = P s 2 S [ x + s ] .Let K bethegroupofcomplex valuedcharactersof K .Givenanadditivecharacter 2 K ,consider [ ]: = P y 2 K y [ y ] and r = P s 2 S s .Wehave A [ ] = r [ ] .Byorthogonalityofcharacters,wemayobservethat f [ ] j 2 K g isabasisof C K andthusthatalleigenvaluesof A areoftheform r forsome additivecharacter .Considerthecharacter 1 denedby 1 x : = e 2 iTr x p .Itisawell-known resultthateverycharacter isoftheform = a ,where a x = 1 ax ,for a 2 K .Let bea generatorof K .Weobservethatforall s 2 S ,wehave r s = r 1 ;andforall b < S [f 0 g ,we 70

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have r b = r .Thustheadjacencymatrixhasatmostthreeeigenvalues k = r 0 , r 1 , r , withgeometricmultiplicities 1 , j S j = k ,and q )]TJ/F71 11.9552 Tf 10.404 0 Td [(k )]TJ/F68 11.9552 Tf 10.405 0 Td [(1 = j S [f 0 gj respectively.Theparameters givenintheTheoremabovecannowbededucedfromthegeneraltheoryofstronglyregular graphs. NowtheeigenvaluesoftheLaplacian L = kI )]TJ/F71 11.9552 Tf 11.041 0 Td [(A are 0 , u : = k )]TJ/F71 11.9552 Tf 10.741 0 Td [(r 1 and v : = k )]TJ/F71 11.9552 Tf 10.741 0 Td [(r , withmultiplicities 0 , k ,and q )]TJ/F71 11.9552 Tf 10.561 0 Td [(k )]TJ/F68 11.9552 Tf 10.562 0 Td [(1 respectively.Wecanseethat v = p q p q + )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 t + 1 ` and u = v + )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 t p q .ItiswellknownthatthenullityoftheLaplacianmatrixofagraphisequal tothenumberofconnectedcomponents.Clearly v , 0 ,andthus G p ;`; t isconnected ifandonlyifeither t iseven,or t isoddand p q , ` )]TJ/F68 11.9552 Tf 10.928 0 Td [(1 .Wewillassumethroughoutthat p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 , ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 whenever t isodd. Givenanelement a inanunramiedextensionofof Q p ,the p -adicvaluationof a is denotedby v p a .Let v p ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 = d ,then v p u = 1 2 ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 t + d and v p v = 1 2 ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 t . ByTheorem 8 : 1 : 2 of[9],wehave L L )]TJ/F68 11.9552 Tf 10.261 0 Td [( v + u I = vuI + J ; where = q )]TJ/F70 11.9552 Tf 10.261 0 Td [(` + 1 + )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 t ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 p q ` 2 .Observingthat LJ = 0 ,weseethattheminimal polynomialof L is x x )]TJ/F71 11.9552 Tf 10.502 0 Td [(u x )]TJ/F71 11.9552 Tf 10.502 0 Td [(v = 0 .Therefore L isdiagonalizable.Asaconsequenceof Kirchhoff'sMatrix-TreeTheoremcf.[29],theorderofcriticalgroupof G p ;`; t is u k v q )]TJ/F71 7.9701 Tf 5.069 0 Td [(k )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 q . 5.2CharacterSumsandBlockDiagonalFormof L . Werecallthedenitionofthegraph G p ;`; t ,itsadjacencymatrix A andLaplaicain matrix L .Thisisthegraphwithvertexset K = F q with q = p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t andedgeset ff x ; y gj x ; y 2 K and x )]TJ/F71 11.9552 Tf 10.696 0 Td [(y 2 S g ,where S isthesubgroupof K withindex ` .By C ,wedenotethecritical groupof G p ;`; t .Wesawin x 5.1that L haseigenvalues 0 , v = p q p q + )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 t + 1 ` and u = v + )]TJ/F68 11.9552 Tf 7.603 0 Td [(1 t p q ,withmultiplicities 1 , q )]TJ/F71 11.9552 Tf 10.26 0 Td [(k )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 and k respectively. Let beaprimitive q )]TJ/F68 11.9552 Tf 10.93 0 Td [(1 -strootofunityinthealgebraicclosureof Q p .Then Q p istheuniqueunramiedextensionofdegree ` )]TJ/F68 11.9552 Tf 10.729 0 Td [(1 t over Q p .Let R betheringofintegers in Q p ,then pR ismaximalin R with R = pR F q = K .Let R K denotethefree R -module 71

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with f [ x ] j x 2 K g asabasisset.Werecallfrom x 3.1,endomorphisms A ; L 2 End R R K , withmatrixrepresentationswithrespectto f [ x ] j x 2 K g A and L respectively.Wehave L [ x ] = k [ x ] )]TJ/F39 11.9552 Tf 12.334 8.216 Td [(P s 2 S [ x + s ] and A [ x ] = P s 2 S [ x + s ] ; x 2 K .As R isanunramiedextension of Z p ,themultiplicityof p j asanelementarydivisorof L isthesameasthatof p j asan elementarydivisoroftheintegermatrix L . Let T : K ! R betheTeichm ullercharactergeneratingthecyclicgroup Hom K ; R . Then K acts R K ,whichdecomposesasthedirectsum R [0] R K .Nowtheregularmodule R K decomposefurtherintoadirectsumof K -invariantsubmodulesofrank 1 ,affordingthe characters T i , i = 0 ;:::; q )]TJ/F68 11.9552 Tf 10.272 0 Td [(2 .Thecomponentaffording T i isspannedby f i : = P x 2 K T i x )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 [ x ] . Therefore f 1 ; f 1 ::: f q )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 ; [0] g isabasisfor R K ,where 1 : = f 0 + [0] = P x 2 K [ x ] . Givenan R -free RS -module M andacharacter : S ! R ,theisotypiccomponentof M correspondingto isthe RS -submodule M : = f m 2 M j sm = s m forall s 2 S g .For 0 < j k )]TJ/F68 11.9552 Tf 10.38 0 Td [(1 ,let N j denotethe R -submoduleof R K with f f j + mk j 0 m ` )]TJ/F68 11.9552 Tf 10.38 0 Td [(1 g asabasisset. Dene N 0 tobethe R -submoduleof R K with f 1 ; [0] ; f k ;::: f ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 k g asabasisset.Then N i isthe isotypiccomponentforthecharacter T i j S ofthegroup S .Wenowhave R K = N 0 N 1 ::: N k )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 : Since S isagroupofautomorphismsfor G p ;`; t ,the R -linearmaps A and L arein fact RS -moduleendomorphisms.Itfollowsthat A and L preservethedecomposition5. For 0 < i k )]TJ/F68 11.9552 Tf 10.378 0 Td [(1 ,let L i denotethematrixof L j N i withrespecttotheorderedbasis f i + mk j 0 m ` )]TJ/F68 11.9552 Tf 10.33 0 Td [(1 .Let L 0 bethematrixof L j N 0 withrespecttotheorderedbasis 1 ; [0] ; f k ;::: f ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 k . Sowithrespecttotheorderedbasis 1 ; [0] ; f k ;::: f ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 k [ k )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 i = 1 f i + mk j 0 m ` )]TJ/F68 11.9552 Tf 11.018 0 Td [(1 ,the matrixrepresentationofthe R -linearmap L is diag L 0 ; L 1 ;:::; L k )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 .Weprovedthefollowing Lemma. Lemma26. As R -matrices, L issimilartotheblockdiagonalmatrix diag L 0 ; L 1 ;:::; L k )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 . Followingconventionsin[1],weextendthe T i 'sto K .Asperthisconvention,the character T 0 mapseveryelementof K to 1 ,while T q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 maps 0 to 0 .Allothercharacters 72

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map 0 to 0 .Fortwointegers a ; b theJacobisum J T a ; T b is P x 2 K T a x T b )]TJ/F71 11.9552 Tf 11.068 0 Td [(x .Wereferthe readertoChapter 2 of[3]forformalpropertiesofJacobisums.Followingtheconventions established,for a . 0mod q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,wehave J T a ; T 0 = 0 and J T a ; T q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 = )]TJ/F68 11.9552 Tf 7.603 0 Td [(1 . ThefollowingLemmadescribesactionof L i on N i . Lemma27. 1.If k i ,wehave L f i = 1 ` qf i )]TJ/F70 7.9701 Tf 11.416 10.607 Td [(` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 1 J T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i ; T )]TJ/F71 7.9701 Tf 5.069 0 Td [(mk f i + mk ! . 2.For 1 j ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,wehave L f jk = 1 ` 1 + qf jk )]TJ/F39 11.9552 Tf 17.851 8.216 Td [(P m , )]TJ/F71 7.9701 Tf 6.265 0 Td [(j ; 0 J T )]TJ/F71 7.9701 Tf 6.264 0 Td [(jk ; T )]TJ/F71 7.9701 Tf 5.069 0 Td [(mk f jk + mk )]TJ/F71 11.9552 Tf 10.26 0 Td [(q [0] ! . 3. L [0] = 1 ` q [0] )]TJ/F70 7.9701 Tf 11.415 10.607 Td [(` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 1 f mk )]TJ/F87 11.9552 Tf 10.26 0 Td [(1 ! . 4. L 1 = 0 . Proof. For x 2 K ,wehave A [ x ] = P y 2 S [ x + y ] . Let S denotethecharacteristicfunctionof S ,treatedasasubsetof K .Wenowhave A [ x ] = P z 2 K S z )]TJ/F71 11.9552 Tf 11.784 0 Td [(x [ z ] .Writing S asalinearcombinationofcharactersof S ,wehave S = 1 ` ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 0 T mk )]TJ/F70 11.9552 Tf 10.26 0 Td [( 0 ! .Here 0 is 1 at 0 and 0 elsewhere. Wehave ` A f i = ` A 0 B B B B B @ X x 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x [ x ] 1 C C C C C A = ` X x 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x X z 2 K S z )]TJ/F71 11.9552 Tf 10.858 0 Td [(x [ z ] = X x 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x X z 2 K T 0 z )]TJ/F71 11.9552 Tf 10.858 0 Td [(x [ z ] + ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 X m = 1 X x 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x X z 2 K T mk z )]TJ/F71 11.9552 Tf 10.858 0 Td [(x [ z ] )]TJ/F39 11.9552 Tf 11.596 12.233 Td [(X x 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x X z 2 K 0 z )]TJ/F71 11.9552 Tf 10.858 0 Td [(x [ z ] : Fromdenitionof f i and 0 ,wehave P x 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x P z 2 K 0 z )]TJ/F71 11.9552 Tf 10.858 0 Td [(x [ z ] = f i . Werecallfromcharactertheorythatforacharacter of K , X x 2 K x = 8 > > > > > > < > > > > > > : q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 if istrivial,and 0 otherwise. 73

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Using4.4,weseethat X x 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x X z 2 K T 0 z )]TJ/F71 11.9552 Tf 10.858 0 Td [(x [ z ] = 0 B B B B B @ X x 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x 1 C C C C C A 0 B B B B B @ X z 2 K [ z ] 1 C C C C C A = 0 Wenowturnourattentionto P x 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x P z 2 K T mk z )]TJ/F71 11.9552 Tf 10.857 0 Td [(x [ z ] ,with 1 m ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 .Wehave X x 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x X z 2 K T mk z )]TJ/F71 11.9552 Tf 10.858 0 Td [(x [ z ] = X x ; z 2 K K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x T mk z )]TJ/F71 11.9552 Tf 10.858 0 Td [(x [ z ] + X x 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x T mk )]TJ/F71 11.9552 Tf 8.202 0 Td [(x [0] For z 2 K ,wehave T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x T mk z )]TJ/F71 11.9552 Tf 11.106 0 Td [(x = T i z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 T mk z T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x = z T mk 1 )]TJ/F68 11.9552 Tf 10.26 0 Td [( x = z .Now,we have X x ; z 2 K K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x T mk z )]TJ/F71 11.9552 Tf 10.857 0 Td [(x [ z ] = X x ; z 2 K K T i z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 T mk z T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x = z T mk 1 )]TJ/F68 11.9552 Tf 10.261 0 Td [( x = z [ z ] = X y ; z 2 K K T i )]TJ/F71 7.9701 Tf 5.069 0 Td [(mk z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i y T mk )]TJ/F71 11.9552 Tf 10.26 0 Td [(y [ z ] = 0 B B B B B B @ X y 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i y T mk )]TJ/F71 11.9552 Tf 10.26 0 Td [(y 1 C C C C C C A 0 B B B B B B @ X z 2 K T i )]TJ/F71 7.9701 Tf 5.069 0 Td [(mk z )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 [ z ] 1 C C C C C C A = J T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i ; T mk f i )]TJ/F71 7.9701 Tf 5.069 0 Td [(mk : Usingtheaboveequalityalongwith5,5,5,wehave ` A f i = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 X m = 0 J T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i ; T mk f i )]TJ/F71 7.9701 Tf 5.069 0 Td [(mk + ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 X m = 1 X x 2 K T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i x T mk )]TJ/F71 11.9552 Tf 8.201 0 Td [(x [0] )]TJ/F71 11.9552 Tf 12.054 0 Td [(f i As )]TJ/F68 11.9552 Tf 7.603 0 Td [(1 2 S and [ K : S ] = k ,wehave T mk )]TJ/F71 11.9552 Tf 8.201 0 Td [(x = T mk x .Thusthemiddlesumaboveis ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 1 P x 2 K T i )]TJ/F71 7.9701 Tf 5.069 0 Td [(mk x ! [0] .Usingthisalongwith4.4in5,weconcludethat aif k i ,wehave ` A f i = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 1 J T )]TJ/F71 7.9701 Tf 5.069 0 Td [(i ; T )]TJ/F71 7.9701 Tf 5.069 0 Td [(mk f i + mk )]TJ/F71 11.9552 Tf 12.053 0 Td [(f i ,and; bfor 1 j ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,wehave ` A f jk = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 1 J T )]TJ/F71 7.9701 Tf 6.265 0 Td [(jk ; T mk f jk )]TJ/F71 7.9701 Tf 5.069 0 Td [(mk + q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1[0] )]TJ/F71 11.9552 Tf 12.054 0 Td [(f jk . Using L = kI )]TJ/F71 11.9552 Tf 11.005 0 Td [(A nowreadilyyields.FromthegeneraltheoryofJacobisums,wehave foranycharacter , J ; )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 = )]TJ/F70 11.9552 Tf 7.604 0 Td [( )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 .Since )]TJ/F68 11.9552 Tf 7.603 0 Td [(1 2 S ,wehave T jk )]TJ/F68 11.9552 Tf 7.603 0 Td [(1 = 1 ,andtherefore wehave J T )]TJ/F71 7.9701 Tf 5.069 0 Td [(mk ; T mk = )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 .Thus ` A f jk = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 1 J T )]TJ/F71 7.9701 Tf 6.264 0 Td [(jk ; T mk f jk )]TJ/F71 7.9701 Tf 5.069 0 Td [(mk + q )]TJ/F68 11.9552 Tf 11.159 0 Td [(1[0] )]TJ/F71 11.9552 Tf 12.952 0 Td [(f jk = P m , )]TJ/F71 7.9701 Tf 6.264 0 Td [(j ; 0 J T )]TJ/F71 7.9701 Tf 6.265 0 Td [(jk ; T mk f jk )]TJ/F71 7.9701 Tf 5.069 0 Td [(mk )]TJ/F87 11.9552 Tf 10.26 0 Td [(1 + q [0] .Nowthefollowsbyusing L = KI )]TJ/F71 11.9552 Tf 10.559 0 Td [(A . Theproofoftheremainingstatementsisstraightforward. 74

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Weobservedin x 5.1that L isdiagonalizableandthussoare L i 's.Werecallfrom x 3that theeigenvaluesof L are 0 , u and v ,withmultiplicities 1 , k and ` )]TJ/F68 11.9552 Tf 10.651 0 Td [(1 k sameas q )]TJ/F71 11.9552 Tf 10.651 0 Td [(k )]TJ/F68 11.9552 Tf 10.651 0 Td [(1 , respectively.Againfrom x 5.1weknowthatthenullityof L is 1 .Nowsincethenullityof L 0 is 1 c.fLemma27,allother L i 'sareinvertible.Itfollowsthatfor i , 0 ,thecharacteristic polynomialof L i isapolynomialoftheform x )]TJ/F71 11.9552 Tf 10.364 0 Td [(u a x )]TJ/F71 11.9552 Tf 10.364 0 Td [(v b with a + b = ` .ByLemma27and diagonalizabilityof L i ,wehave q = tr L i = au + bv .Itnowfollowsthat a = 1 and b = ` )]TJ/F68 11.9552 Tf 10.515 0 Td [(1 . Bysimilararguments,wemayshowthattheeigenvaluesof L 0 are 0 , u and v withgeometric multiplicities 1 , 1 and ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,respectively.WehaveprovedthefollowingLemma. Lemma28. 1.For i , 0 ,theeigenvaluesof L i are u and v withgeometricmultiplicities 1 and ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,respectively. 2.Theeigenvaluesof L 0 are 0 , u and v withgeometricmultiplicities 1 , 1 and ` )]TJ/F68 11.9552 Tf 11.379 0 Td [(1 , respectively. 5.3TheSylow p -SubgroupoftheCriticalGroupof G p ;`; t ByLemma26,itisclearthatndingtheelementarydivisorsof R -matrices L i 'swill determinethe p -elementarydivisorsofthecriticalgroup.As ` isaunitin R ,theSmithnormal formof L i isthesameasthatof ` L i .Lemma27showsthattheanyentryof ` L i iseither q or isaJacobisumoftheform J T )]TJ/F68 7.9701 Tf 5.069 0 Td [( i + mk ; T )]TJ/F71 7.9701 Tf 5.069 0 Td [(nk ,where 0 m ` )]TJ/F68 11.9552 Tf 10.589 0 Td [(1 and 0 < n ` )]TJ/F68 11.9552 Tf 10.59 0 Td [(1 .Inthe contextofLemma1,itisworthinvestigatingthe p -adicvaluationsofJacobisums. Aninteger a notdivisibleby q )]TJ/F68 11.9552 Tf 10.918 0 Td [(1 has,whenreducedmodulo q )]TJ/F68 11.9552 Tf 10.918 0 Td [(1 ,aunique p -digit expansion a a 0 + a 1 p + ::: + a ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 mod q )]TJ/F68 11.9552 Tf 9.565 0 Td [(1 ,where 0 a i p )]TJ/F68 11.9552 Tf 9.566 0 Td [(1 .Werepresent thisexpansionbythetupleofdigits a 0 ;:::; a i ;:::; a ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 .By s a wedenotethesum P a i . Forexample, 1 hastheexpansion ;:::; 0 ;::: 0 and s = 1 . ApplyingStickelberger'stheoremonGaussSums[30]andthewellknowrelation betweenGaussandJacobisumswecandeducethefollowingtheorem. 75

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Theorem29. Let q beapowerofaprime p andlet a and b beintegersnotdivisibleby q )]TJ/F68 11.9552 Tf 10.317 0 Td [(1 . If a + b . 0mod q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,thenwehave v p J T )]TJ/F71 7.9701 Tf 5.069 0 Td [(a ; T )]TJ/F71 7.9701 Tf 5.069 0 Td [(b = s a + s b )]TJ/F71 11.9552 Tf 10.858 0 Td [(s a + b p )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 : Inotherwords,the p -adicvaluationof J T )]TJ/F71 7.9701 Tf 5.069 0 Td [(a ; T )]TJ/F71 7.9701 Tf 5.069 0 Td [(b isequaltothenumberofcarries,when adding p -expansionsof a and b modulo q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . Given b 2 Z ,by [ b ] denotetheuniquepositiveintegerlessthan ` satisfying b [ b ] mod ` .Wecannowseethat k = q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ` = p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ` p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 X i = 0 [ p i ] p )]TJ/F68 11.9552 Tf 10.26 0 Td [([ p i + 1 ] ` ! p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F71 7.9701 Tf 5.069 0 Td [(i t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 X i = 0 p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 i : Thusinthenotationweadopted,thetuplefor k isthetupleinwhichthestring [ p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 ] p )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 ` ;:::; [ p i ] p )]TJ/F68 11.9552 Tf 10.26 0 Td [([ p i + 1 ] ` ;:::; p )]TJ/F68 11.9552 Tf 10.261 0 Td [([ p ] ` ! repeats t times.As p isprimitivemodulo ` ,wehave f [ p i ] j 0 i ` )]TJ/F68 11.9552 Tf 10.431 0 Td [(2 g = f 1 ; 2 ;:::` )]TJ/F68 11.9552 Tf 10.431 0 Td [(1 g .We cannowconcludethat s k = t ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 P i = 0 [ p i ] p )]TJ/F68 7.9701 Tf 5.069 0 Td [([ p i + 1 ] ` = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 p )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . Nowfor 0 i ; j ` )]TJ/F68 11.9552 Tf 10.473 0 Td [(1 ,wecannd r i ; j 2 Z suchthat [ p i ][ p j ] = [ p i + j ] + r i ; j ` .Givenany m 2f 1 ; 2 ;:::;` )]TJ/F68 11.9552 Tf 10.307 0 Td [(1 g ,as p isprimitive mod ` ,thereisapositiveinteger j suchthat [ p j ] = m . 76

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Wehave mk = [ p j ] p ` )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ` p ` )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 p ` )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 .Nowwehave [ p j ] p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ` ! = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 X i = 0 [ p j ] [ p i ] p )]TJ/F68 11.9552 Tf 10.261 0 Td [([ p i + 1 ] ` ! p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F71 7.9701 Tf 5.069 0 Td [(i = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 X i = 0 [ p i + j ] p + pr i ; j ` )]TJ/F68 11.9552 Tf 10.26 0 Td [([ p i + 1 + j ] )]TJ/F71 11.9552 Tf 10.26 0 Td [(r i + 1 ; j ` ` ! p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F71 7.9701 Tf 5.069 0 Td [(i = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 X i = 0 [ p i + j ] p )]TJ/F68 11.9552 Tf 10.26 0 Td [([ p i + 1 + j ] ` ! p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F71 7.9701 Tf 5.069 0 Td [(i + ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 X i = 0 r i ; j p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F71 7.9701 Tf 5.069 0 Td [(i )]TJ/F70 7.9701 Tf 11.92 14.624 Td [(` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 X i = 0 r i + 1 ; j p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F71 7.9701 Tf 5.069 0 Td [(i = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 X i = 0 [ p i + j ] p )]TJ/F68 11.9552 Tf 10.26 0 Td [([ p i + 1 + j ] ` ! p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F71 7.9701 Tf 5.069 0 Td [(i + r 0 ; j p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F71 11.9552 Tf 10.26 0 Td [(r ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; j : As r 0 ; j = r ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 ; j = 0 ,fromtheabovecomputationweobservethat [ p j ] p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ` ! p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 p ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 X i = 0 [ p i + j ] p )]TJ/F68 11.9552 Tf 10.26 0 Td [([ p i + 1 + j ] ` ! p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 )]TJ/F71 7.9701 Tf 5.069 0 Td [(i t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 X i = 0 p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 i Thusinthenotationweadopted,thetuplefor [ p j ] k isthetupleinwhichthestring [ p ` + j )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 ] p )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ` ;:::; [ p i + j ] p )]TJ/F68 11.9552 Tf 10.26 0 Td [([ p i + j + 1 ] ` ;:::; [ p j ] p )]TJ/F68 11.9552 Tf 10.26 0 Td [([ p j + 1 ] ` ! repeats t times.Sothedigitsof mk = [ p j ] k canbeobtainedbypermutingthedigitsof k ,and thus s mk = s k = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 p )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . Given a ; b asdescribedinthetheoremabove,by c a ; b wedenote v p J T )]TJ/F71 7.9701 Tf 5.069 0 Td [(a ; T )]TJ/F71 7.9701 Tf 5.069 0 Td [(b .Then byLemma27theoff-diagonalentriesof L i with i , 0 are u mn p c i + mk ; nk forsomeunits u mn of R ,andthediagonalentriesareall q =` .Lemma28showsthat L i satises x )]TJ/F71 11.9552 Tf 10.535 0 Td [(u x )]TJ/F71 11.9552 Tf 10.534 0 Td [(v = 0 . Wemakeuseofthistoarriveatthefollowinglemma. Lemma30. Given j < ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 and 0 < i k )]TJ/F68 11.9552 Tf 10.467 0 Td [(1 ,themultiplicityof p j asanelementarydivisor of L i isthesameasthatof p v p uv )]TJ/F71 7.9701 Tf 6.265 0 Td [(j . Proof. As L i satises x )]TJ/F71 11.9552 Tf 10.973 0 Td [(u x )]TJ/F71 11.9552 Tf 10.973 0 Td [(v = 0 ,wehave L i L i )]TJ/F68 11.9552 Tf 10.973 0 Td [( v + u I = vuI .Let P and Q be R -matricessuchthat PL i Q istheSmithnormalformof L i .Nowconsider PL i QQ )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 L i )]TJ/F68 11.9552 Tf -440.834 -23.907 Td [( v + u I P )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 = vuI .Thisshowsthatthemultiplicityof p v p uv )]TJ/F71 7.9701 Tf 6.265 0 Td [(j asanelementarydivisorof L i isthesameasthemultiplicityof p j asanelementarydivisorof L i )]TJ/F68 11.9552 Tf 10.788 0 Td [( v + u I .Since L i and 77

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L i )]TJ/F68 11.9552 Tf 10.348 0 Td [( u + v I arecongruentmodulo p v p v = p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 ,for 0 j < ` )]TJ/F68 11.9552 Tf 10.347 0 Td [(1 t = 2 ,themultiplicityof p j asanelementarydivisorof L i )]TJ/F68 11.9552 Tf 10.273 0 Td [( v + u I isthesameasthemultiplicityof p j asanelementary divisorof L i . WenowcomputetheSmithnormalformsof L i 's. Lemma31. iFor 0 < i ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,theSmithnormalformof L i over R isthediagonalmatrix diag p min i ; p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 ;:::; p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 | {z } ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 repetitions ; p v p uv )]TJ/F51 7.9701 Tf 5.069 0 Td [(min i . Here min i = min f c i + mk ; nk j 0 m ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 and 0 < n ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 g . iiTheSmithnormalformof L 0 over R is diag ; 1 ; p ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 | {z } ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(3 times ; p v p u ; 0 . Proof. Givenan R -matrix X ,by X ,wedenotethe p -adicvaluationoftheproductofa completesetofnon-zeroinvariantfactorsof X ,countedwithmultiplicities.Bythenotationin x 2.1, e i X denotesthemultiplicityof p i asanelementarydivisorof X .Followingthenotation in x 2.1,weconsiderthevectorspaces M y X with y 2 Z 0 .WeuseLemma3toproveour results. AsaconsequenceofKirchhoff'sMatrix-Treetheorem,wehave L = v p j C j = v p u k v q )]TJ/F71 5.9776 Tf 3.802 0 Td [(k )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 q = kv p u + ` )]TJ/F68 11.9552 Tf 11.213 0 Td [(1 kv p v )]TJ/F71 11.9552 Tf 11.213 0 Td [(v p q .Lemma28impliesthatfor i , 0 ,wehave L i = v p det L i = v p u + ` )]TJ/F68 11.9552 Tf 11.359 0 Td [(1 v p v .ApplicationofLemma26givesus L 0 = L )]TJ/F39 11.9552 Tf 10.26 8.217 Td [(P i , 0 L i = v p u + ` )]TJ/F68 11.9552 Tf 10.26 0 Td [(3 v p v . iByTheorem29,wehave c i + mk ; nk + c i + m + n k ; ` )]TJ/F71 11.9552 Tf 11.067 0 Td [(n k = ` )]TJ/F68 11.9552 Tf 11.067 0 Td [(1 t .We cannowconcludethat min i ` )]TJ/F68 11.9552 Tf 10.715 0 Td [(1 t = 2 .Let diag 1 ; 2 ;::: ` betheSmithnormalform of L i .ThenbyLemma27andLemma1,itfollowsthat min i = v p 1 .Bydenitionof M min i L i itfollowsthat M min i L i = N i andthus dim M min i L i = ` .Firstweassumethat min i < ` )]TJ/F68 11.9552 Tf 8.629 0 Td [(1 t = 2 .Inthiscase,byLemma30wehave e v p uv )]TJ/F51 7.9701 Tf 5.069 0 Td [(min i L i = e min i L i 1 andthus dim M v p uv )]TJ/F51 7.9701 Tf 5.069 0 Td [(min i L i 1 .Lemma28tellusthatgeometricmultiplicityof v asaneigenvalue of L i is ` )]TJ/F68 11.9552 Tf 10.555 0 Td [(1 .NowLemma4impliesthat dim M ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 L i ` )]TJ/F68 11.9552 Tf 10.555 0 Td [(1 .Applying2,wehave dim M min i L i )]TJ/F68 11.9552 Tf 10.678 0 Td [(dim M ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 L i = P min i j < ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 e j L i 1 ,andthus dim M min i L i ` . ThereforebyLemma3,setting j = 3 , s 1 = `; s 2 = ` )]TJ/F68 11.9552 Tf 11.392 0 Td [(1 ; s 3 = 1 ; s 4 = ker L i = 0 , 78

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t 1 = min i ; t 2 = ` )]TJ/F68 11.9552 Tf 10.557 0 Td [(1 t = 2 ,and t 3 = v p uv )]TJ/F51 11.9552 Tf 10.556 0 Td [(min i ,wehave e min i L i = e v p uv )]TJ/F51 7.9701 Tf 5.069 0 Td [(min i L i = 1 , e ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 L i = ` )]TJ/F68 11.9552 Tf 11.108 0 Td [(2 ,and e i L i = 0 forallother i .Nowassumethat min i = ` )]TJ/F68 11.9552 Tf 11.108 0 Td [(1 t = 2 . Lemma4impliesthat dim M v p u L i 1 ,since u isaneigenvalueofmultiplicity 1 .Therefore byLemma3,setting j = 2 , s 1 = ` , s 2 = 1 , s 3 = ker L i , t 1 = min i = ` )]TJ/F68 11.9552 Tf 11.119 0 Td [(1 t = 2 ,and t 2 = v p u = v p uv )]TJ/F68 11.9552 Tf 10.372 0 Td [( ` )]TJ/F68 11.9552 Tf 10.372 0 Td [(1 t = 2 ,wehave e ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t = 2 L i = ` )]TJ/F68 11.9552 Tf 10.373 0 Td [(1 , e v p uv )]TJ/F51 7.9701 Tf 5.069 0 Td [(min i L i = 1 ,and e i L i = 0 forallother i .Thuswehavei. iiByLemma27andTheorem29,thereareunits v mn in R suchthatthematrix ` L 0 is 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 qv p q ::: v ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 p q )]TJ/F68 11.9552 Tf 7.603 0 Td [(10 : : : : : : ::: : : : : : : : : : v ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(11 p q :::::: q )]TJ/F68 11.9552 Tf 7.603 0 Td [(10 )]TJ/F71 11.9552 Tf 7.604 0 Td [(q :::::: )]TJ/F71 11.9552 Tf 7.603 0 Td [(q q 0 1 :::::: 1 )]TJ/F68 11.9552 Tf 7.603 0 Td [(10 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : Thedeterminantofthe 2 2 minor 2 6 6 6 6 6 6 6 6 6 4 q )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 1 )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 3 7 7 7 7 7 7 7 7 7 5 of ` L 0 isaunitin R .Observethatany 3 3 minorof ` L 0 has p -valuationofatleast v p q .NowapplyingLemma1yieldsthatthe multiplicityof p 0 = 1 asanelementarydivisorof L 0 is 2 ,thatis e 0 L 0 = 2 .NowLemma2 impliesthat dim M 0 L 0 )]TJ/F68 11.9552 Tf 9.511 0 Td [(dim M 1 L 0 = 2 ,andthuswehave dim M 1 L 0 = ` + 1 )]TJ/F68 11.9552 Tf 9.511 0 Td [(2 = ` )]TJ/F68 11.9552 Tf 9.511 0 Td [(1 . ByLemma28andLemma4,wehave dim M v p v L 0 ` )]TJ/F68 11.9552 Tf 10.409 0 Td [(1 .Since M 1 L 0 M v p v L 0 ,we have dim M v p v L 0 = ` )]TJ/F68 11.9552 Tf 10.436 0 Td [(1 .Lemma27impliesthat im L 0 isgeneratedby 1 and P j , 0 f jk + 1 . Therefore dim im L 0 = 2 .As LJ = 0 ,by5therestrictionof L to im L satises L L )]TJ/F71 11.9552 Tf 9.513 0 Td [(v + uI = vuI .As im L 0 Im L ,wecanconcludethat im L 0 M v p uv L 0 M v p u L 0 . Wehave L 0 = v p v ` )]TJ/F68 11.9552 Tf 10.679 0 Td [(1 )]TJ/F68 11.9552 Tf 10.68 0 Td [(2 + v p u )]TJ/F68 11.9552 Tf 10.679 0 Td [(1 = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 ` )]TJ/F68 11.9552 Tf 10.679 0 Td [(1 )]TJ/F68 11.9552 Tf 10.679 0 Td [(2 + v p u )]TJ/F68 11.9552 Tf 10.679 0 Td [(1 .Now applicationofLemma3yieldsii. 5.4ProofofTheorem15 Lemma26showstheLaplacianmatrix L issimilarover R totheblockdiagonalmatrix diag L 0 ; L 1 ; L 2 ::: L k )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 : 79

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Results , , ,and nowfollowbyapplyingLemma31. If p ` )]TJ/F68 11.9552 Tf 10.552 0 Td [(1 ,wehave d = v p ` )]TJ/F68 11.9552 Tf 10.552 0 Td [(1 = 0 and v p u = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 = v p v = v p uv )]TJ/F68 7.9701 Tf 11.747 5.491 Td [( ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 .From 2,wehave q )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 = P j , ` )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 t 2 e j + e ` )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 t 2 .Nowapplicationof and yields . If p j ` )]TJ/F68 11.9552 Tf 10.826 0 Td [(1 ,then v p u > ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 = v p v and v p u = v p uv )]TJ/F68 7.9701 Tf 12.021 5.491 Td [( ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 .NowbyLemma26 andLemma31,wededucethat e v p u = jf i j 1 i k )]TJ/F68 11.9552 Tf 10.701 0 Td [(1 and min i = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 gj + 1 .Wemay alsodeducethat e ` )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 t 2 = ` )]TJ/F68 11.9552 Tf 10.861 0 Td [(2 jf i j 1 i k )]TJ/F68 11.9552 Tf 10.861 0 Td [(1 and min i = ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 gj + ` )]TJ/F68 11.9552 Tf 10.861 0 Td [(3 jf i j 1 i k )]TJ/F68 11.9552 Tf 10.55 0 Td [(1 and c i < ` )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 t 2 gj + ` )]TJ/F68 11.9552 Tf 10.549 0 Td [(3 andthusthat e v p u )]TJ/F68 11.9552 Tf 10.549 0 Td [(1 + ` )]TJ/F68 11.9552 Tf 10.549 0 Td [(2 k )]TJ/F68 11.9552 Tf 10.549 0 Td [(1 + ` )]TJ/F68 11.9552 Tf 10.549 0 Td [(3 = e ` )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 t 2 .From 2,wehave q )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 = e v p u + e ` )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 t 2 + P j < f ` )]TJ/F68 5.9776 Tf 3.802 0 Td [(1 t 2 ; v p u g e j .Nowapplicationof and yield . 5.5TheCriticalGroupof G p ; 3 ; t Wenowturnourfocustographsoftheform G p ; 3 ; t .Weassumethat p ; t , ; 1 and p 2mod3 ,sothesegraphsareconnectedandstronglyregular.Recallthatthisis theCayleygraphontheadditivegroupoftheeld K = F q q = p 2 t with“connection”set S , where S istheuniquesubgroupof K satisfying k : = j S j = q )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 3 .Alltheresultsintheprevious sectionstransfertothiscasebysetting ` = 3 . Lemma31showsthatfor i , 0 ,theSmithnormalformof L i over R isthematrix diag p c ; p t ; p v p uv )]TJ/F71 7.9701 Tf 5.069 0 Td [(c .Here c istheleastamongthe p -adicvaluationsoftheentriesof L i .In thiscase, v = p q p q + )]TJ/F68 11.9552 Tf 7.603 0 Td [(1 t + 1 3 and u = v + )]TJ/F68 11.9552 Tf 7.603 0 Td [(1 t p q arethenon-zeroeigenvaluesofthe Laplacianof G p ; 3 ; t . Givenintegers a ; b notdivisibleby q )]TJ/F68 11.9552 Tf 10.669 0 Td [(1 ,let c a ; b denotethenumberofcarrieswhen addingthe p -adicexpansionsof a and b mod q )]TJ/F68 11.9552 Tf 11.21 0 Td [(1 .Considerthefollowingcounting problem. 5.5.0.0.1CountingProblem: .For 1 i k )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 ,by min i wedenote min f c i + mk ; nk j 0 m 2 ; and n = 1 ; 2 g : Given 0 a < t ,nd jf i j min i = a gj . 80

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Givenapositiveinteger a ,by e a wedenotethemultiplicityof p a asanelementarydivisor ofthecriticalgroupof G p ; 3 ; t .Let e 0 bethe p -rankoftheLaplacianof G p ; 3 ; t .Theorem 15impliesthat,for 0 < a < t ,wehave e a = jf i j min i = a gj ,and e 0 = jf i j min i = 0 gj + 2 .Thus thesolutiontothisproblemwillimmediatelyprovideuswiththe p -elementarydivisorsofthe criticalgroupsofgraphsoftheform G p ; 3 ; t . Everyinteger a thatisnotdivisibleby q )]TJ/F68 11.9552 Tf 11.506 0 Td [(1 ,whenreducedmodulo q )]TJ/F68 11.9552 Tf 11.506 0 Td [(1 ,hasa unique p -adicexpansion a 2 t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 0 a m p m mod q )]TJ/F68 11.9552 Tf 10.963 0 Td [(1 ,where 0 a m p )]TJ/F68 11.9552 Tf 10.963 0 Td [(1 .By s a ,we denote P a m .The p -adicexpressionfor k is t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 0 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 3 p 2 m + 2 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 3 p 2 m + 1 andthatof 2 k is t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 0 2 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 3 p 2 m + p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 3 p 2 m + 1 .Thuswehave s k = s k = t p )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . WemayobservefromTheorem29that c a ; b = s a + s b )]TJ/F71 11.9552 Tf 10.858 0 Td [(s a + b p )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 : Given j 2 Z ,by j wedenotetheuniqueelementof f 0 ; 1 ;:::; q )]TJ/F68 11.9552 Tf 11.018 0 Td [(2 g satisfying j j mod q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 . Thefollowingfollowsfrom5. Lemma32. Givenaninteger j . 0mod q )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 and m = )]TJ/F68 11.9552 Tf 7.604 0 Td [(1 ; 1 ,thefollowinghold. 1. c j ; mk + c j + mk ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(mk = 2 t 2. c j ; mk + c j + mk ; mk = t + c j ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(mk 3. c j ; mk = c )]TJ/F71 11.9552 Tf 9.396 0 Td [(j )]TJ/F71 11.9552 Tf 10.26 0 Td [(mk ; mk Let j 2f 1 ;:::; q )]TJ/F68 11.9552 Tf 9.631 0 Td [(2 gnf k ; 2 k g ,dene g j : = f c j ; k ; c j ; 2 k g .Forevery j ,thereisaunique j 2f 1 ; 2 :::; k )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 g suchthat j )]TJ/F70 11.9552 Tf 10.261 0 Td [( j 2f 0 ; k ; 2 k g .Notethat )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 i = f i ; i + k ; i + 2 k g . For 0 a t ,wedene Y a : = f j j g j = f a ; b g forsome b suchthat a b t g and R a = f i j 1 i k )]TJ/F68 11.9552 Tf 10.263 0 Td [(1 and min i = a g .FromTheorem15,wehavei e a = j R a j ,for 0 < a < t and; ii e 0 = j R 0 j + 2 . Lemma33. Given Y a and denedaboveand a < t ,thefollowingaretrue. 1.If a istherestrictionof to Y a ,then a Y a = R a . 2.Let i 2 R a .If j 2 )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 a i and m j 2f 1 ; 2 g suchthat c j ; m j k = a ,then 81

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a j + m j k < )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 a i b )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 a i = f j g ifandonlyif t < g j ; cand )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 a i = f j ; j )]TJ/F71 11.9552 Tf 10.26 0 Td [(m j k g ifandonlyif t 2 g j . 3.For 0 a < t ,wehave j R a j = j Y a j)]TJ/F68 7.9701 Tf 16.504 4.711 Td [(1 2 jf j j g j = f a ; t ggj = j Y a j)-222(jf j j g j = f a ggj Proof. 1Let m 2f 1 ; 2 g and j 2 Y a suchthat c j ; mk = a and c j ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(mk = b .Thenby Lemma32,wehave f c j + mk ; nk j 0 m 2 ; and n = 1 ; 2 g = f a ; b ; t )]TJ/F71 11.9552 Tf 11.43 0 Td [(a + b ; t )]TJ/F71 11.9552 Tf -418.411 -23.908 Td [(b )]TJ/F71 11.9552 Tf 11.116 0 Td [(a ; 2 t )]TJ/F71 11.9552 Tf 11.116 0 Td [(a ; 2 t )]TJ/F71 11.9552 Tf 11.116 0 Td [(b g .Since a b t ,wehave min j = a andthus a Y a R a .If i 2 R a ,thenthereexists j 2f i ; i + k ; i + 2 k g and m 2 1 ; 2 suchthat c j ; mk = a .Using a = min f c i + mk ; nk j 0 m 2 ; and n = 1 ; 2 g ,andLemma32,wehave c j ; mk c j )]TJ/F71 11.9552 Tf 10.26 0 Td [(mk ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(mk = t + c j ; mk )]TJ/F71 11.9552 Tf 10.31 0 Td [(c j ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(mk .Thuswehave c j ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(mk t andtherefore j 2 Y a and a j = i . 2FromLemma32wehave c j ; m j k + c j + m j k ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(m j k = 2 t and c j ; m j k + c j + m j k ; m j k = t + c j ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(m j k .As c j ; m j k = a < t ,Lemma32implies c j + m j k ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(m j k = 2 t )]TJ/F71 11.9552 Tf 11.592 0 Td [(a > t .As j 2 Y a ,wehave c j ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(m j k c j ; m j k andthus c j + m j k ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(m j k = t + c j ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(m j k )]TJ/F71 11.9552 Tf 10.26 0 Td [(c j ; m j k t .Thus j + m j k < )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 a i .Wehave )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 a i f j ; j )]TJ/F71 11.9552 Tf 10.26 0 Td [(mk g As j 2 Y a ,wehavethat a = c j ; m j K c j ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(m j k t .Now,applicationofLemma32 yields c j )]TJ/F71 11.9552 Tf 10.261 0 Td [(m j k ; m j k = 2 t )]TJ/F71 11.9552 Tf 7.981 0 Td [(c j ; mk t .ByLemma32,wehave c j )]TJ/F71 11.9552 Tf 10.26 0 Td [(m j k ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(m j k + c j ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(m j k = t + c j ; m j k ,andthus c j )]TJ/F71 11.9552 Tf 10.26 0 Td [(m j k ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(m j k = a ifandonlyif c j ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(m j k = t .Therefore j )]TJ/F71 11.9552 Tf 10.26 0 Td [(m j k 2 Y a isandonlyif c j ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(m j k = t .Thusistrue. 3Fromand,wehave j R a j = P i 2 Y a i j )]TJ/F68 5.9776 Tf 3.801 0 Td [(1 a i j = j Y a j)]TJ/F68 7.9701 Tf 18.818 4.71 Td [(1 2 jf j j g j = f a ; t ggj .Given j 2f j j g j = f a ; t gg ,let m j 2f 1 ; 2 g suchthat c j ; m j k = a .Wethenhave c j ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(m j k = t .By Lemma32,wehave c j ; m j k + t = c j ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(m j k + c j )]TJ/F71 11.9552 Tf 10.26 0 Td [(m j k ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(m j k andthus c j )]TJ/F71 11.9552 Tf 10.26 0 Td [(m j k ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(m j k = c j ; m j k = a .Using c j ; mk = c )]TJ/F71 11.9552 Tf 9.397 0 Td [(j )]TJ/F71 11.9552 Tf 10.26 0 Td [(mk ; mk and c )]TJ/F71 11.9552 Tf 9.396 0 Td [(j )]TJ/F71 11.9552 Tf 10.26 0 Td [(mk ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(mk = c j )]TJ/F71 11.9552 Tf 10.26 0 Td [(mk ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(mk from Lemma32,wehave c )]TJ/F71 11.9552 Tf 9.397 0 Td [(j )]TJ/F71 11.9552 Tf 10.26 0 Td [(m j k ; m j k = c j ; m j k = c j )]TJ/F71 11.9552 Tf 10.26 0 Td [(m j k ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(m j k = c )]TJ/F71 11.9552 Tf 9.396 0 Td [(j )]TJ/F71 11.9552 Tf 10.26 0 Td [(m j k ; )]TJ/F71 11.9552 Tf 7.604 0 Td [(m j k . Thusif j 2f j j g j = f a ; t gg ,then )]TJ/F71 11.9552 Tf 9.396 0 Td [(j )]TJ/F71 11.9552 Tf 10.261 0 Td [(m j k 2f j j g j = f a gg .Nowthemap : f j j g j = f a ; t gg!f j j g j = f a gg denedby j = )]TJ/F71 11.9552 Tf 9.397 0 Td [(j )]TJ/F71 11.9552 Tf 10.26 0 Td [(m j k iswell-dened.For j 2f j j g j = f a gg , wehave )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 j f )]TJ/F71 11.9552 Tf 9.396 0 Td [(j ; )]TJ/F71 11.9552 Tf 9.397 0 Td [(j + k ; )]TJ/F71 11.9552 Tf 9.397 0 Td [(j + 2 k g .Given m 2f 1 ; 2 g ,Lemma32givesus c )]TJ/F71 11.9552 Tf 9.397 0 Td [(j ; mk = 82

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c j )]TJ/F71 11.9552 Tf 10.26 0 Td [(mk ; mk = 2 t )]TJ/F71 11.9552 Tf 8.476 0 Td [(c j ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(mk = 2 t )]TJ/F71 11.9552 Tf 8.476 0 Td [(a , c )]TJ/F71 11.9552 Tf 9.397 0 Td [(j + mk ; mk = c j + mk ; mk = t + c j ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(mk )]TJ/F71 11.9552 Tf 8.476 0 Td [(c j ; mk = t + a )]TJ/F71 11.9552 Tf 10.923 0 Td [(a = t ,and c )]TJ/F71 11.9552 Tf 9.397 0 Td [(j + mk ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(mk = c j ; )]TJ/F71 11.9552 Tf 7.603 0 Td [(mk = a .Thusthemap isa 2 to 1 mapand therefore 1 2 jf j j g j = f a ; t ggj = jf j j g j = f a ggj . Corollary34. e 0 = p + 1 3 ! 2 t t + 1 )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 . Proof. Lemma33andTheorem15imply e 0 = j R 0 j + 2 = j Y 0 j)-252(f j j g j = f 0 gg + 2 .Werecall that k = t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 0 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 3 p 2 m + 2 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 3 p 2 m + 1 and 2 k = t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 0 2 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 3 p 2 m + p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 3 p 2 m + 1 .Thereforeset f j j c j ; k ; c j ; 2 k = ; b g ismadeupofnumbersoftheform j = t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 P m = 0 a 2 m p 2 m + a m + 1 p 2 m + 1 satisfying:i 0 a 2 m < p + 1 3 ,ii 0 a 2 m + 1 < 2 p + 1 3 and,iii j < f k ; 2 k g .Thusthissethassize 2 t p + 1 3 ! 2 t )]TJ/F68 11.9552 Tf 8.27 0 Td [(2 .Similarargumentsyeild jf j j j , 0 and c j ; k ; c j ; 2 k = b ; 0 gj = 2 t p + 1 3 ! 2 t )]TJ/F68 11.9552 Tf 8.27 0 Td [(2 and jf j j j , 0 and g j = f 0 ggj = p + 1 3 2 t )]TJ/F68 11.9552 Tf 10.275 0 Td [(1 .Theresultnowfollowsbytheprincipleofinclusionexclusion. For 0 < a < t ,Lemma33showsthat e a = j R a j .Wewillusethetransfermatrixmethodto compute j R a j .Weconstructaweighteddigraph D ,andchangetheproblemofcomputing e a tothatofcountingclosedwalksin D ofcertainlengthandweight. Let D beadigraphwithvertexset V ,edgeset E ,andwithaweightfunction wt : E ! R withvaluesinsomecommutativering R .By M ,wedenotetheadjacencymatrixof D with respecttotheweight wt .Given n 2 Z > 0 ,let C n = P wt ,wherethesumisoverclosed walksin D oflength n .ThefollowingLemmawhichisCorollary 4 : 7 : 3 of[28]givesusthe generatingfunction P n 1 C n z n . Lemma35. Let T z = det I )]TJ/F71 11.9552 Tf 10.26 0 Td [(zM ,then P n 1 C n z n = )]TJ/F71 11.9552 Tf 8.799 8.094 Td [(zT 0 z T z . Consider A 1 = f ;; j ;; 2f 0 ; 1 ;:::; p )]TJ/F68 11.9552 Tf 11.553 0 Td [(1 gf 1 ; 2 gf 1 ; 2 gg and A 2 = f [ ;; ] j ;; 2f 0 ; 1 ;:::; p )]TJ/F68 11.9552 Tf 11.312 0 Td [(1 gf 1 ; 2 gf 1 ; 2 gg .Weconstructabipartitedigraph 83

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D = A 1 [ A 2 ; E .Thereisanarc e 2 E from ;; 2 A 1 to [ 0 ; 0 ; 0 ] 2 A 2 ifanonlyif + 2 p )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 3 + = + p 0 and + p )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 3 + = + p 0 forsome ; 2f 0 ; 1 ;:::; p )]TJ/F68 11.9552 Tf 10.444 0 Td [(1 g .Thereisanarc e 2 E from [ ;; ] 2 A 2 to 0 ; 0 ; 0 2 A 1 if andonlyif + p )]TJ/F68 11.9552 Tf 10.261 0 Td [(2 3 + = + p 0 and + 2 p )]TJ/F68 11.9552 Tf 10.261 0 Td [(1 3 + = + p 0 forsome ; 2f 0 ; 1 ;:::; p )]TJ/F68 11.9552 Tf 11.001 0 Td [(1 g .Thearcsin D oftype e and e areassignedlabel and weights wt e = wt e = x 0 y 0 .Sowehaveaweightfunction wt : E ! C [ x ; y ] on D .The weightofawalkon D willbetheproductsoftheweightsofitsarcs. Given a ; b 2f 0 ; 1 ; 2 ;:::; 2 t + 1 g ,let E a ; b bethesetofclosedwalksoflength 2 t andweight x a y b .Aclosedwalkoflength 2 t withitsinitialvertexin A 1 issaidtobeoftype A 1 ,andisof type A 2 otherwise.Let Y a ; b = f j 2f 1 ; 2 ;:::; q )]TJ/F68 11.9552 Tf 10.565 0 Td [(2 gnf k ; 2 k gj g j = f a ; b gg .Let a 0 ; a 1 ;::: a 2 t be thelabelsofarcsofawalk w 2[ E a ; b ,thendene w = P a i p i .When f a ; b gf 0 ; 2 t g = ; , wehave E a ; b Y a ; b .Bythe p -aryadd-with-carry-algorithmdescribedinTheorem4.1 of[16],given j 2 Y a ; b ,thereexist carrysequences 0 ; 1 ;::: 2 t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 and 0 ; 1 ;::: 2 t )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 with i ; i 2f 1 ; 2 g suchthat a i + 2 p )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 3 + i = b i + i + 1 pa i + p )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 3 + i = d i + i + 1 p ; foreven i and; a i + p )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 3 + i = b i + i + 1 pa i + 2 p )]TJ/F68 11.9552 Tf 10.26 0 Td [(1 3 + i = d i + i + 1 p ; forodd i : Here j = P a i p i , j + k = P b i p i and j + 2 k = d i p i .Wecannowseethatthereare exactlytwoclosedwalks,oneofeachtypewhichmapto j under .If w j ; A 1 respectively 84

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w j ; A 2 isthewalkoftype A 1 respectivelytype A 2 suchthat w j ; A 1 = j respectively w j ; A 2 = j ,then wt w j ; A 1 = x c j ; k y c j ; 2 k respectively wt w j ; A 2 = x c j ; 2 k y c j ; k .We cannowconcludethatfor a , b ,therestrictionof isabijectionfrom E a ; b to Y a ; b .Applying Lemma333givesus e a = t X b = a + 1 j E a ; b j ; forall 0 < a < t . Weobservethatforall ; 0 2f 0 ; 1 ;:::; p )]TJ/F68 11.9552 Tf 10.683 0 Td [(1 g and ; 2f 1 ; 2 gf 1 ; 2 g ,thereisnoarc from ;; resp. [ ;; ] to [ 0 ; 0 ; 1] resp. 0 ; 1 ; 0 .Wemayalsoconcludethat 1.thereisanedgefrom ;; to [ 0 ; 0 ; 0] ifandonlyif 0 < p + 1 3 )]TJ/F70 11.9552 Tf 10.26 0 Td [( ; 2.thereisanedgefrom ;; to [ 0 ; 1 ; 0] ifandonlyif p + 1 3 )]TJ/F70 11.9552 Tf 10.26 0 Td [( < 2 p + 1 3 )]TJ/F70 11.9552 Tf 10.261 0 Td [( ; 3.thereisanedgefrom ;; to [ 0 ; 1 ; 1] ifandonlyif 2 p + 1 3 )]TJ/F70 11.9552 Tf 10.26 0 Td [( < p ; 4.thereisanedgefrom [ ;; ] to 0 ; 0 ; 0 ifandonlyif 0 < p + 1 3 )]TJ/F70 11.9552 Tf 10.26 0 Td [( ; 5.thereisanedgefrom [ ;; ] to 0 ; 0 ; 1 ifandonlyif p + 1 3 )]TJ/F70 11.9552 Tf 10.26 0 Td [( < 2 p + 1 3 )]TJ/F70 11.9552 Tf 10.261 0 Td [( ; 6.andthereisanedgefrom [ ;; ] to 0 ; 1 ; 1 ifandonlyif 2 p + 1 3 )]TJ/F70 11.9552 Tf 10.26 0 Td [( < p . Let M betheadjacencymatrixoftheweighteddigraph D andlet U bethe C x ; y vector spacegeneratedbythevertexset A 1 [ A 2 of D asabasis.Byabuseofnotation,wemay assume M 2 End U . Let h 1 : = P ; P 0 < p + 1 3 )]TJ/F70 7.9701 Tf 5.069 0 Td [( [ 0 ;; ] , h 2 : = P ; P p + 1 3 )]TJ/F70 7.9701 Tf 5.069 0 Td [( 0 < 2 p + 1 3 )]TJ/F70 7.9701 Tf 5.069 0 Td [( [ 0 ;; ] , h 3 : = P ; P 0 2 p + 1 3 )]TJ/F70 7.9701 Tf 5.069 0 Td [( [ 0 ;; ] , h 0 1 : = P ; P 0 < p + 1 3 )]TJ/F70 7.9701 Tf 5.069 0 Td [( 0 ;; , h 0 2 = P ; P p + 1 3 )]TJ/F70 7.9701 Tf 5.069 0 Td [( 0 < 2 p + 1 3 )]TJ/F70 7.9701 Tf 5.069 0 Td [( 0 ;; ,and h 0 3 : = P ; P 0 2 p + 1 3 )]TJ/F70 7.9701 Tf 5.069 0 Td [( 0 ;; . Wecanseethat M A 1 [ A 2 = f h 1 ; h 2 ; h 3 ; h 0 1 ; h 0 2 ; h 0 3 g .Wealsohave, 85

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M h 0 1 = p + 1 3 h 1 + p + 1 3 xh 2 + p )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 3 xyh 3 ; M h 0 2 = p + 1 3 h 1 + p )]TJ/F68 11.9552 Tf 10.261 0 Td [(2 3 xh 2 + p + 1 3 xyh 3 ; M h 0 3 = p )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 3 h 1 + p + 1 3 xh 2 + p + 1 3 xyh 3 ; M h 1 = p + 1 3 h 0 1 + p + 1 3 yh 0 2 + p )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 3 xyh 0 3 ; M h 2 = p + 1 3 h 0 1 + p )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 3 yh 0 2 + p + 1 3 xyh 0 3 ; and M h 3 = p )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 3 h 0 1 + p + 1 3 yh 0 2 + p + 1 3 xyh 0 3 : Let W bethesubspaceof U generatedby f h i ; h 0 i j i = 1 ; 2 ; 3 g ,then M U = W .The set = f h i ; h 0 i j i = 1 ; 2 ; 3 g islinearlyindependent,andthusisabasisfor W .Let M [ ] bethe matrixrepresentationof M j W withrespecttothebasis of W .Fromaboveweseethat M [ ] is 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 000 p + 1 3 p + 1 3 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 3 000 p + 1 3 y p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 3 y p + 1 3 y 000 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 y xy p + 1 3 xy p + 1 3 xy p + 1 3 p + 1 3 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 3 000 p + 1 3 x p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 3 x p + 1 3 x 000 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 y xy p + 1 3 xy p + 1 3 xy 000 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : As det M [ ] = )]TJ/F71 11.9552 Tf 8.5 0 Td [(p 2 x 3 y 3 , 0 ,wehave W ker M = f 0 g andthus U = ker M W . Thusthecharacteristicpolynomialof M is f z = z 8 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(6 det zI )]TJ/F71 11.9552 Tf 10.861 0 Td [(M [ ] .Carefulcomputation showsthat det zI )]TJ/F71 11.9552 Tf 10.858 0 Td [(M [ ] = z 6 )]TJ/F71 11.9552 Tf 10.559 0 Td [(Pz 4 + Qz 2 )]TJ/F71 11.9552 Tf 10.261 0 Td [(R ; where P = p + 1 3 2 x 2 y 2 + x 2 y + xy 2 + x + y + 1 + p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 3 2 3 xy , Q = p + 1 3 2 xy x 2 y 2 + x 2 y + xy 2 + x + y + 1 + 2 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(1 3 2 3 x 2 y 2 ,and R = p 2 x 3 y 3 : Thus f z = z 8 p )]TJ/F71 11.9552 Tf 10.559 0 Td [(Pz 8 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + Qz 8 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(4 )]TJ/F71 11.9552 Tf 10.26 0 Td [(Rz 8 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(6 . 86

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Let C n = P wt ,wherethesumisoverclosedwalksin D oflength n .As D isa bipartitegraph,wehave C n = 0 forallodd n .ByLemma35,wehave X t 1 C t z 2 t = )]TJ/F71 11.9552 Tf 8.799 8.094 Td [(zT 0 z T z ; where T z = det I )]TJ/F71 11.9552 Tf 10.79 0 Td [(zM .Thecharacteristicpolynomialof M wascomputedabovetobe z 8 p )]TJ/F71 11.9552 Tf 10.559 0 Td [(Pz 8 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(2 + Qz 8 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(4 )]TJ/F71 11.9552 Tf 10.26 0 Td [(Rz 8 p )]TJ/F68 7.9701 Tf 5.069 0 Td [(6 ,andthuswehave X t 1 C t z 2 t = 2 Pz 2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(4 Qz 4 + 6 Rz 6 1 )]TJ/F68 11.9552 Tf 10.26 0 Td [( Pz 2 )]TJ/F71 11.9552 Tf 10.559 0 Td [(Qz 4 + Rz 6 : Let C t = 0 for t 0 .Wehave P t 1 C t )]TJ/F71 11.9552 Tf 10.924 0 Td [(PC t )]TJ/F68 11.9552 Tf 10.626 0 Td [(2 + QC t )]TJ/F68 11.9552 Tf 10.626 0 Td [(4 )]TJ/F71 11.9552 Tf 10.626 0 Td [(RC t )]TJ/F68 11.9552 Tf 10.625 0 Td [(6 z t = 2 Pz )]TJ/F68 11.9552 Tf 10.26 0 Td [(4 Qz 2 + 6 Rz 3 .Thuswehave C = 2 P C = 2 P 2 )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 Q ; C = 6 R + 2 P 3 )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 QP )]TJ/F68 11.9552 Tf 10.261 0 Td [(2 PQ ; and C t = PC t )]TJ/F68 11.9552 Tf 10.26 0 Td [(2 )]TJ/F71 11.9552 Tf 10.559 0 Td [(QC t )]TJ/F68 11.9552 Tf 10.26 0 Td [(4 + RC t )]TJ/F68 11.9552 Tf 10.26 0 Td [(6 for t > 3 : Thecoefcientof x a y b in C t is j E a ; b j .Given a < t ,wehavefrom5that e a = P a < b t j E a ; b j .ApplicationofTheorem15andCorollary34yieldsTheorem16. 87

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REFERENCES [1]JamesAx.Zeroesofpolynomialsoverniteelds. AmericanJournalofMathematics , 86:255,1964. [2]HuaBai.Onthecriticalgroupofthen-cube. LinearAlgebraanditsApplications , 369:251–261,2003. [3]BruceCBerndt,KennethSWilliams,andRonaldJEvans. GaussandJacobisums . Wiley,1998. [4]N.L.Biggs.Chip-ringandthecriticalgroupofagraph. JournalofAlgebraicCombinatorics ,9:25,Jan1999. [5]AndersBjorner,LaszloLovasz,andPeterW.Shor.Chip-ringgamesongraphs. EuropeanJournalofCombinatorics ,12:283–291,1991. [6]R.C.Bose.Stronglyregulargraphs,partialgeometriesandpartiallybalanceddesigns. PacicJ.Math. ,13:389,1963. [7]AndriesBrouwer,JoshuaDucey,andPeterSin.Theelementarydivisorsofthe incidencematrixofskewlinesin PG ; q . ProceedingsoftheAmericanMathematical Society ,140:2561,2012. [8]AndriesE.Brouwer,ArjehM.Cohen,andArnoldNeumaier. Distanceregulargraphs . Springer,1989. [9]AndriesE.BrouwerandWillemH.Haemers. Spectraofgraphs .Universitext.Springer, NewYork,2012. [10]DavidB.Chandler,PeterSin,andQingXiang.TheSmithandcriticalgroupsofPaley graphs. JournalofAlgebraicCombinatorics ,41:1013,Jun2015. [11]DeepakDhar.Self-organizedcriticalstateofsandpileautomatonmodels. Physical ReviewLetters ,64:1613,1990. [12]JoshuaEDucey,JonathanGerhard,andNoahWatson.TheSmithandcritical groupsofthesquarerook'sgraphanditscomplement. TheElectronicJournalof Combinatorics ,23:4,2016. [13]JoshuaEDucey,IanHill,andPeterSin.ThecriticalgroupoftheKnesergraphon 2-subsetsofann-elementset. LinearAlgebraanditsApplications ,546:154,2018. [14]JoshuaEDuceyandPeterSin.TheSmithgroupandthecriticalgroupoftheGrassmanngraphoflinesinniteprojectivespaceandofitscomplement. arXivpreprint arXiv:1706.01294 ,2017. [15]C.GodsilandG.F.Royle. AlgebraicGraphTheory .GraduateTextsinMathematics. SpringerNewYork,2001. 88

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[16]TorHelleseth,HenkDLHollmann,AlexanderKholosha,ZeyingWang,andQingXiang. Proofsoftwoconjecturesonternaryweaklyregularbentfunctions. IEEETransactions onInformationTheory ,55:5272,2009. [17]BrianJacobson,AndrewNiedermaier,andVictorReiner.Criticalgroupsforcomplete multipartitegraphsandcartesianproductsofcompletegraphs. JournalofGraph Theory ,44:231,2003. [18]EricS.Lander. SymmetricDesigns:AnAlgebraicApproach .LondonMathematical SocietyLectureNoteSeries.CambridgeUniversityPress,1983. [19]JMLataille,PeterSin,andPhamHuuTiep.Themodulo2structureofrank3permutationmodulesforoddcharacteristicsymplecticgroups. JournalofAlgebra , 268:463,2003. [20]MartinWLiebeck.Permutationmodulesforrank3unitarygroups. JournalofAlgebra , 88:317,1984. [21]MartinWLiebeck.Permutationmodulesforrank3symplecticandorthogonalgroups. JournalofAlgebra ,92:9,1985. [22]DinoLorenzini.SmithnormalformandLaplacians. JournalofCombinatorialTheory, SeriesB ,98:1271–1300,2008. [23]MorrisNewman. Integralmatrices ,volume45.AcademicPress,1972. [24]VenkataRaghuTejPantangi.CriticalgroupsofvanLint-SchrijverCyclotomicStrongly RegularGraphs. arXive-prints ,pagearXiv:1810.01003,October2018. [25]VenkataRaghuTejPantangiandPeterSin.SmithandcriticalgroupsofPolargraphs. arXivpreprintarXiv:1706.08175 ,2017. [26]PeterSin.ThecriticalgroupsofthePeisertgraphs. JournalofAlgebraicCombinatorics , 48:227,2018. [27]PeterSinandPhamHuuTiep.Rank3permutationmodulesoftheniteclassical groups. JournalofAlgebra ,291:551,2005. [28]RichardP.Stanley. EnumerativeCombinatorics:Volume1 .CambridgeUniversityPress, NewYork,NY,USA,2ndedition,2011. [29]RichardPStanley.Smithnormalformincombinatorics. JournalofCombinatorial Theory,SeriesA ,144:476,2016. [30]L.Stickelberger.UebereineVerallgemeinerungderKreistheilung. Mathematische Annalen ,37:321,Sep1890. [31]JacobusHvanLintandAlexanderSchrijver.Constructionofstronglyregulargraphs, two-weightcodesandpartialgeometriesbyniteelds. Combinatorica ,1:63, 1981. 89

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BIOGRAPHICALSKETCH VenkataRaghuTejPantangiwasbornin1990inHyderabad,India.Hegraduatedwith aB.S-M.SdualdegreefromIndianInstituteofScienceEducationandResearch,Punein 2012.In2019hereceivedadoctorateinmathematicsfromUniversityofFlorida,underthe guidanceofPeterSin. 90