Zeta Functions of Graphs: A Stroll through the Garden

253 Pages·2010·2.34 MB·English

by  Audrey Terras

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CAMBRIDGE STUDIES IN ADVANCED MATHEMATICS 128
EditorialBoard
B. BOLLOBA´S, W. FULTON, A. KATOK, F. KIRWAN,
P. SARNAK, B. SIMON, B. TOTARO
ZETA FUNCTIONS OF GRAPHS
Graph theory meets number theory in this stimulating book. Ihara zeta functions of
finitegraphsarereciprocalsofpolynomials,sometimesinseveralvariables.Analogies
aboundwithnumber-theoreticfunctionssuchasRiemannorDedekindzetafunctions.
Forexample,thereisaRiemannhypothesis(whichmaybefalse)andaprimenumber
theorem for graphs. Explicit constructions of graph coverings use Galois theory to
generalizeCayleyandSchreiergraphs.Thennon-isomorphicsimplegraphswiththe
samezetafunctionareproduced,showingthatyoucannot“hear”theshapeofagraph.
The spectra of matrices such as the adjacency and edge adjacency matrices of a
graph are essential to the plot of this book, which makes connections with quantum
chaos and random matrix theory and also with expander and Ramanujan graphs, of
interestincomputerscience.Pitchedatbeginninggraduatestudents,thebookwillalso
appeal to researchers. Many well-chosen illustrations and exercises, both theoretical
andcomputer-based,areincludedthroughout.
Audrey Terras is Professor Emerita of Mathematics at the University of California,
SanDiego.
CAMBRIDGESTUDIESINADVANCEDMATHEMATICS
EditorialBoard:
B.Bolloba´s,W.Fulton,A.Katok,F.Kirwan,P.Sarnak,B.Simon,B.Totaro
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Zeta Functions of Graphs
A Stroll through the Garden
AUDREY TERRAS
UniversityofCalifornia,SanDiego
CAMBRIDGEUNIVERSITYPRESS
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(cid:2)c A.Terras2011
Thispublicationisincopyright.Subjecttostatutoryexception
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thewrittenpermissionofCambridgeUniversityPress.
Firstpublished2011
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
AcatalogrecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloging-in-PublicationData
Terras,Audrey.
Zetafunctionsofgraphs:astrollthroughthegarden/AudreyTerras.
p. cm.–(Cambridgestudiesinadvancedmathematics;128)
ISBN978-0-521-11367-0(Hardback)
1. Graphtheory. 2. Functions,Zeta. I. Title. II. Series.
QA166.T472010
(cid:3)
511.5–dc22
2010024611
ISBN 978-0-521-11367-0Hardback
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Contents
Listofillustrations pageviii
Preface xi
PartI Aquicklookatvariouszetafunctions 1
1 Riemannzetafunctionandotherzetasfrom
numbertheory 3
2 Iharazetafunction 10
2.1 Theusualhypothesesandsomedefinitions 10
2.2 Primesin X 11
2.3 Iharazetafunction 12
2.4 Fundamentalgroupofagraphanditsconnection
withprimes 13
2.5 Iharadeterminantformula 17
2.6 Coveringgraphs 20
2.7 Graphtheoryprimenumbertheorem 21
3 Selbergzetafunction 22
4 Ruellezetafunction 27
5 Chaos 31
PartII Iharazetafunctionandthegraphtheoryprime
numbertheorem 43
6 Iharazetafunctionofaweightedgraph 45
v
vi Contents
7 Regulargraphs,locationofpolesoftheIharazeta,
functionalequations 47
8 Irregulargraphs:whatistheRiemannhypothesis? 52
9 DiscussionofregularRamanujangraphs 61
9.1 Randomwalksonregulargraphs 61
9.2 Examples:thePaleygraph,two-dimensionalEuclidean
graphs,andthegraphsofLubotzky,Phillips,andSarnak 63
9.3 WhytheRamanujanboundisbestpossible
(AlonandBoppanatheorem) 68
9.4 WhyareRamanujangraphsgoodexpanders? 70
9.5 WhydoRamanujangraphshavesmalldiameters? 73
10 Graphtheoryprimenumbertheorem 75
10.1 WhichgraphpropertiesaredeterminedbytheIharazeta? 78
PartIII Edgeandpathzetafunctions 81
11 Edgezetafunctions 83
11.1 DefinitionsandBass’sproofoftheIharathree-term
determinantformula 83
11.2 PropertiesofW andaproofofthetheoremofKotani
1
andSunada 90
12 Pathzetafunctions 98
PartIV FiniteunramifiedGaloiscoveringsofconnected
graphs 103
13 FiniteunramifiedcoveringsandGaloisgroups 105
13.1 Definitions 105
13.2 Examplesofcoverings 111
13.3 Someramificationexperiments 115
14 FundamentaltheoremofGaloistheory 117
15 Behaviorofprimesincoverings 128
16 Frobeniusautomorphisms 133
17 Howtoconstructintermediatecoveringsusingthe
Frobeniusautomorphism 141
Contents vii
18 Artin L-functions 144
18.1 Briefsurveyonrepresentationsoffinitegroups 144
18.2 DefinitionoftheArtin–Ihara L-function 148
18.3 PropertiesofArtin–Ihara L-functions 154
18.4 ExamplesoffactorizationsofArtin–Ihara L-functions 157
19 EdgeArtin L-functions 164
19.1 DefinitionandpropertiesofedgeArtin L-functions 164
19.2 Proofsofdeterminantformulasforedge
Artin L-functions 169
19.3 Proofoftheinductionproperty 173
20 PathArtin L-functions 178
20.1 DefinitionandpropertiesofpathArtin L-functions 178
20.2 Inductionproperty 180
21 Non-isomorphicregulargraphswithoutloopsormultiedges
havingthesameIharazetafunction 186
22 Chebotarevdensitytheorem 194
23 Siegelpoles 200
23.1 SummaryofSiegelpoleresults 200
23.2 ProofofTheorems23.3and23.5 202
23.3 Generalcase;inflationanddeflation 206
PartV Lastlookatthegarden 209
24 Anapplicationtoerror-correctingcodes 211
25 Explicitformulas 216
26 Againchaos 218
27 Finalresearchproblems 227
References 230
Index 236
Illustrations
1.1 GraphofthemodulusoftheRiemannzeta 5
1.2 Factsaboutzetafunctionsand L-functions 6
1.3 Whatzetaand L-functionssayaboutnumberfields 6
1.4 Statisticsofprimeidealsandzeros 7
1.5 Splittingofprimesinquadraticextensions 8
2.1 “Bad”graphforthetheoryofzetafunctions 11
2.2 Anarbitraryorientationoftheedgesofagraph 11
2.3 Bouquetofloops 14
2.4 Partofthe4-regulartree 16
2.5 Tetrahedrongraph K and K −e 16
4 4
2.6 ContourmapofthemodulusofthereciprocalofIharazeta
for K intheu-variable 19
4
2.7 ContourmapofthemodulusofreciprocalofIharazetafor
K inthes-variable 19
4
2.8 Thecubeasaquadraticcoveringofthetetrahedron 21
3.1 FailureofEuclid’sfifthpostulate 23
3.2 Fundamentaldomainfor H modSL(2,Z) 23
3.3 Tessellationofupperhalfplanefromthemodulargroup 24
3.4 Imagesofpointsontwogeodesiccirclesaftermappinginto
fundamentaldomainthenbyCayleytransformintotheunitdisc 25
5.1 Spectraandlevelspacings 32
5.2 Spectraof200randomreal50×50symmetricmatrices 33
5.3 Spectrumofrandomnormalreal1001×1001matrix 35
5.4 Levelspacinghistogramsfor166Erandanucleardataensemble 36
5.5 Odlyzko’scomparisonofthelevelspacingsofzerosofthe
RiemannzetafunctionandthosefortheGUE 37
viii