Zeta Functions of Graphs: A Stroll through

Graph theory meets number theory in this stimulating book. Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann/Dedekind zeta functions. For example, there is a Riemann hypothesis (which may be false) and prime number theorem for graphs. Explicit constructions of graph coverings use Galois theory to generalize Cayley and Schreier graphs. Then non-isomorphic simple graphs with the same zeta are produced, showing you cannot hear the shape of a graph. 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, plus expander/Ramanujan graphs of interest in computer science. Pitched at beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and diagrams, and exercises throughout, theoretical and computer-based.

List of illustrations	p. viii
Preface	p. xi
A quick look at various zeta functions	p. 1
Riemann zeta function and other zetas from number theory	p. 3
Ihara zeta function	p. 10
The usual hypotheses and some definitions	p. 10
Primes in X	p. 11
Ihara zeta function	p. 12
Fundamental group of a graph and its connection with primes	p. 13
Ihara determinant formula	p. 17
Covering graphs	p. 20
Graph theory prime number theorem	p. 21
Selberg zeta function	p. 22
Ruelle zeta function	p. 27
Chaos	p. 31
Ihara zeta function and the graph theory prime number theorem	p. 43
Ihara zeta function of a weighted graph	p. 45
Regular graphs, location of poles of the Ihara zeta, functional equations	p. 47
Irregular graphs: what is the Riemann hypothesis?	p. 52
Discussion of regular Ramanujan graphs	p. 61
Random walks on regular graphs	p. 61
Examples: the Paley graph, two-dimensional Euclidean graphs, and the graphs of Lubotzky, Phillips, and Sarnak	p. 63
Why the Ramanujan bound is best possible (Alon and Boppana theorem)	p. 68
Why are Ramanujan graphs good expanders?	p. 70
Why do Ramanujan graphs have small diameters?	p. 73
Graph theory prime number theorem	p. 75
Which graph properties are determined by the Ihara zeta?	p. 78
Edge and path zeta functions	p. 81
Edge zeta functions	p. 83
Definitions and Bass's proof of the Ihara three-term determinant formula	p. 83
Properties of W₁ and a proof of the theorem of Kotani and Sunada	p. 90
Path zeta functions	p. 98
Finite unramified Galois coverings of connected graphs	p. 103
Finite unramified coverings and Galois groups	p. 105
Definitions	p. 105
Examples of coverings	p. 111
Some ramification experiments	p. 115
Fundamental theorem of Galois theory	p. 117
Behavior of primes in coverings	p. 128
Frobenius automorphisms	p. 133
How to construct intermediate coverings using the Frobenius automorphism	p. 141
Artin L-functions	p. 144
Brief survey on representations of finite groups	p. 144
Definition of the Artin-Ihara L-function	p. 148
Properties of Artin-Ihara L-functions	p. 154
Examples of factorizations of Artin-Ihara L-functions	p. 157
Edge Artin L-functions	p. 164
Definition and properties of edge Artin L-functions	p. 164
Proofs of determinant formulas for edge Artin L-functions	p. 169
Proof of the induction property	p. 173
Path Artin L-functions	p. 178
Definition and properties of path Artin L-functions	p. 178
Induction property	p. 180
Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function	p. 186
Chebotarev density theorem	p. 194
Siegel poles	p. 200
Summary of Siegel pole results	p. 200
Proof of Theorems 23.3 and 23.5	p. 202
General case; inflation and deflation	p. 206
Last look at the garden	p. 209
An application to error-correcting codes	p. 211
Explicit formulas	p. 216
Again chaos	p. 218
Final research problems	p. 227
References	p. 230
Index	p. 236
Table of Contents provided by Ingram. All Rights Reserved.

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Zeta Functions of Graphs: A Stroll through the Garden

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