Zeta Functions of Graphs: A Stroll through the Garden

by Audrey Terras

Overview

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.

Product Details

ISBN-13:	9780521113670
Publisher:	Cambridge University Press
Publication date:	11/18/2010
Series:	Cambridge Studies in Advanced Mathematics , #128
Pages:	252
Product dimensions:	6.10(w) x 9.00(h) x 0.80(d)

About the Author

Audrey Terras is Professor of Mathematics at the University of California, San Diego.

Table of Contents

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.

Editorial Reviews

"The book brings together existing research into a coherent presentation..."
Christopher Knox Storm, Mathematical Reviews

From the Publisher