Zeta Functions of Graphs
Zeta Functions of Graphs
$82.99
USDGraph 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.
- Makes connections with quantum chaos and random matrix theory, plus Ramanujan graphs, which are of interest to computer scientists
- Explains key ideas using lots of well-chosen illustrations, alongside theoretical and computer-based exercises
- Perfect for beginning graduate students, or established researchers who want a stimulating introduction to the topic
Reviews & endorsements
'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
Product details
November 2010Hardback
9780521113670
252 pages
235 × 157 × 19 mm
0.53kg
65 b/w illus. 11 colour illus. 95 exercises
Available
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.
