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.
Author(s): Audrey Terras
Series: Cambridge Studies in Advanced Mathematics 128
Edition: 1
Publisher: Cambridge University Press
Year: 2010
Language: English
Pages: 253
Cover......Page 1
Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 7
Illustrations......Page 10
Preface......Page 13
I A quick look at various zeta functions......Page 15
1 Riemann zeta function and other zetas from number theory......Page 17
2.1 The usual hypotheses and some definitions......Page 24
2.2 Primes in X......Page 25
2.3 Ihara zeta function......Page 26
2.4 Fundamental group of a graph and its connection with primes......Page 27
2.5 Ihara determinant formula......Page 31
2.6 Covering graphs......Page 34
2.7 Graph theory prime number theorem......Page 35
3 Selberg zeta function......Page 36
4 Ruelle zeta function......Page 41
5 Chaos......Page 45
II Ihara zeta function and the graph theory prime number theorem......Page 57
6 Ihara zeta function of a weighted graph......Page 59
7 Regular graphs, location of poles of the Ihara zeta, functional equations......Page 61
8 Irregular graphs: what is the Riemann hypothesis?......Page 66
9.1 Random walks on regular graphs......Page 75
9.2 Examples: the Paley graph, two-dimensional Euclidean graphs, and the graphs of Lubotzky, Phillips, and Sarnak......Page 77
9.3 Why the Ramanujan bound is best possible (Alon and Boppana theorem)......Page 82
9.4 Why are Ramanujan graphs good expanders?......Page 84
9.5 Why do Ramanujan graphs have small diameters?......Page 87
10 Graph theory prime number theorem......Page 89
10.1 Which graph properties are determined by the Ihara zeta?......Page 92
III Edge and path zeta functions......Page 95
11.1 Definitions and Bass's proof of the Ihara three-term determinant formula......Page 97
11.2 Properties of W1 and a proof of the theorem of Kotaniand Sunada......Page 104
12 Path zeta functions......Page 112
IV Finite unramified Galois coverings of connected graphs......Page 117
13.1 Definitions......Page 119
13.2 Examples of coverings......Page 125
13.3 Some ramification experiments......Page 129
14 Fundamental theorem of Galois theory......Page 131
15 Behavior of primes in coverings......Page 142
16 Frobenius automorphisms......Page 147
17 How to construct intermediate coverings using the Frobenius automorphism......Page 155
18.1 Brief survey on representations of finite groups......Page 158
18.2 Definition of the Artin–Ihara L-function......Page 162
18.3 Properties of Artin–Ihara L-functions......Page 168
18.4 Examples of factorizations of Artin–Ihara L-functions......Page 171
19.1 Definition and properties of edge Artin L-functions......Page 178
19.2 Proofs of determinant formulas for edge Artin L-functions......Page 183
19.2.1 Bass proof of Ihara theorem for Artin L-functions......Page 184
19.3 Proof of the induction property......Page 187
20.1 Definition and properties of path Artin L-functions......Page 192
20.2 Induction property......Page 194
21 Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function......Page 200
22 Chebotarev density theorem......Page 208
23.1 Summary of Siegel pole results......Page 214
23.2 Proof of Theorems 23.3 and 23.5......Page 216
23.3 General case inflation and deflation......Page 220
V Last look at the garden......Page 223
24 An application to error-correcting codes......Page 225
25 Explicit formulas......Page 230
26 Again chaos......Page 232
27 Final research problems......Page 241
References......Page 244
Index......Page 250
