Zeta Functions of Graphs

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

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

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.