Zeta Functions of Graphs
A Stroll through the Garden

Combinatorics meets number theory in this stimulating stroll through the zetas. Includes well-chosen illustrations and exercises, both theoretical and computer-based.

Audrey Terras (Author)

9780521113670, Cambridge University Press

Hardback, published 18 November 2010

252 pages, 65 b/w illus. 11 colour illus. 95 exercises
23.5 x 15.7 x 1.9 cm, 0.53 kg

'The book is very appealing through its informal style and the variety of topics covered and may be considered the standard reference book in this field.' Zentralblatt MATH

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. Created for beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and exercises, both theoretical and computer-based, are included throughout.

List of illustrations
Preface
Part I. A Quick Look at Various Zeta Functions: 1. Riemann's zeta function and other zetas from number theory
2. Ihara's zeta function
3. Selberg's zeta function
4. Ruelle's zeta function
5. Chaos
Part II. Ihara's Zeta Function and the Graph Theory Prime Number Theorem: 6. Ihara zeta function of a weighted graph
7. Regular graphs, location of poles of zeta, functional equations
8. Irregular graphs: what is the RH?
9. Discussion of regular Ramanujan graphs
10. The graph theory prime number theorem
Part III. Edge and Path Zeta Functions: 11. The edge zeta function
12. Path zeta functions
Part IV. Finite Unramified Galois Coverings of Connected Graphs: 13. Finite unramified coverings and Galois groups
14. Fundamental theorem of Galois theory
15. Behavior of primes in coverings
16. Frobenius automorphisms
17. How to construct intermediate coverings using the Frobenius automorphism
18. Artin L-functions
19. Edge Artin L-functions
20. Path Artin L-functions
21. Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function
22. The Chebotarev Density Theorem
23. Siegel poles
Part V. Last Look at the Garden: 24. An application to error-correcting codes
25. Explicit formulas
26. Again chaos
27. Final research problems
References
Index.

Subject Areas: Number theory [PBH], Discrete mathematics [PBD]