Holdings: Zeta functions of graphs

Zeta functions of graphs: a stroll through the garden

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Bibliographic Details
Main Author:	Terras, Audrey 1942- (Author)
Format:	Book
Language:	English
Published:	Cambridge [u.a.] Cambridge Univ. Press 2011
Edition:	1. publ.
Series:	Cambridge studies in advanced mathematics 128
Subjects:	Zetafunktion
Online Access:	Inhaltsverzeichnis
Physical Description:	XII, 239 S. Ill., graph. Darst.
ISBN:	9780521113670

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MARC


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adam_text	Contents List of illustrations page viii Preface xi Part I A quick look at various zeta functions 1 1 Riemann zeta function and other zetas from number theory 3 Ihara zeta function 10 2.1 The usual hypotheses and some definitions 10 2.2 Primes in X 11 2.3 Ihara zeta function 12 2.4 Fundamental group of a graph and its connection with primes 13 2.5 Ihara determinant formula 17 2.6 Covering graphs 20 2.7 Graph theory prime number theorem 21 3 Selberg zeta function 22 4 Ruelle zeta function 27 5 Chaos 31 Part II Ihara zeta function and the graph theory prime number theorem 43 6 Ihara zeta function of a weighted graph 45 vi Contents 7 Regular graphs, location of poles of the Ihara zeta, functional equations 47 8 Irregular graphs: what is the Riemann hypothesis? 52 9 Discussion of regular Ramanujan graphs 61 9.1 Random walks on regular graphs 61 9.2 Examples: the Paley graph, two-dimensional Euclidean graphs, and the graphs of Lubotzky, Phillips, and Sárnak 63 9.3 Why the Ramanujan bound is best possible (Alon and Boppana theorem) 68 9.4 Why are Ramanujan graphs good expanders? 70 9.5 Why do Ramanujan graphs have small diameters? 73 10 Graph theory prime number theorem 75 10.1 Which graph properties are determined by the Ihara zeta? 78 Partili Edge and path zeta functions 81 11 Edge zeta functions 83 11.1 Definitions and Bass's proof of the Ihara three-term determinant formula 83 11.2 Properties of W\ and a proof of the theorem of Kotani and Sunada 90 12 Path zeta functions 98 Part IV Finite unramified Galois coverings of connected graphs 103 13 Finite unramified coverings and Galois groups 105 13.1 Definitions 105 13.2 Examples of coverings 111 13.3 Some ramification experiments 115 14 Fundamental theorem of Galois theory 117 15 Behavior of primes in coverings 128 16 Frobenius automorphisms 133 17 How to construct intermediate coverings using the Frobenius automorphism 141 Contents vii 18 Artin ¿-functions 144 18.1 Brief survey on representations of finite groups 144 18.2 Definition of the Artin-Ihara L-function 148 18.3 Properties of Artin-Ihara L -functions 154 18.4 Examples of factorizations of Artin-Ihara L -functions 157 19 Edge Artin L-fimctions 164 19.1 Definition and properties of edge Artin L -functions 164 19.2 Proofs of determinant formulas for edge Artin L -functions 169 19.3 Proof of the induction property 173 20 Path Artin L-functions 178 20.1 Definition and properties of path Artin L-functions 178 20.2 Induction property 180 21 Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function 186 22 Chebotarev density theorem 194 23 Siegel poles 200 23.1 Summary of Siegel pole results 200 23.2 Proof of Theorems 23.3 and 23.5 202 23.3 General case; inflation and deflation 206 Part V Last look at the garden 209 24 An application to error-correcting codes 211 25 Explicit formulas 216 26 Again chaos 218 27 Final research problems 227 References 230 Index 236
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isbn	9780521113670
language	English
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spelling	Terras, Audrey 1942- Verfasser (DE-588)115217320 aut Zeta functions of graphs a stroll through the garden Audrey Terras 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2011 XII, 239 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge studies in advanced mathematics 128 Zetafunktion (DE-588)4190764-4 gnd rswk-swf Zetafunktion (DE-588)4190764-4 s DE-604 Cambridge studies in advanced mathematics 128 (DE-604)BV000003678 128 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020690832&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis
spellingShingle	Terras, Audrey 1942- Zeta functions of graphs a stroll through the garden Cambridge studies in advanced mathematics Zetafunktion (DE-588)4190764-4 gnd
subject_GND	(DE-588)4190764-4
title	Zeta functions of graphs a stroll through the garden
title_auth	Zeta functions of graphs a stroll through the garden
title_exact_search	Zeta functions of graphs a stroll through the garden
title_full	Zeta functions of graphs a stroll through the garden Audrey Terras
title_fullStr	Zeta functions of graphs a stroll through the garden Audrey Terras
title_full_unstemmed	Zeta functions of graphs a stroll through the garden Audrey Terras
title_short	Zeta functions of graphs
title_sort	zeta functions of graphs a stroll through the garden
title_sub	a stroll through the garden
topic	Zetafunktion (DE-588)4190764-4 gnd
topic_facet	Zetafunktion
url	http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020690832&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA
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