Zeta Functions of Graphs: A Stroll through the Garden (Hardback)

Condition

VERY GOOD: This book is in very good condition, showing only slight signs of use and wear. There is still a new book "crackle" when it is opened to be read.

Product Details

This is #128 in theCambridge studies in advanced mathematics series from Cambridge University Press.

From the back cover: "Ihara zeta functions of finite graphs are reciprocals of polynomials, sometimes in several variables. Analogies abound with number-theoretic functions such as Riemann or Dedekind zeta functions. For example, there is a Riemann hypothesis (which may be false) and a 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 function are produced, showing that 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, and also with expander and Ramanujan graphs, of interest in computer science. Pitched at beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and exercises, both theoretical and computer-based, are included throughout."

BRIEF CONTENTS

Part I. A quick look at various zeta functions

1. Riemann zeta functions and other zetas from number theory
2. Ihara zeta functions
3. Selberg zeta functions
4. Ruelle zeta functions
5. Chaos

Part II. Ihara zeta function and the graph theory prime number theorem

6. Ihara zeta function of a weighted graph
7. Regular graphs, location of poles of the Ihara zeta, functional equations
8. Irregular graphs: what is the Riemann hypothesis?
9. Discussion of regular Ramanujan graphs
10. Graph theory prime number theorem

Part III. Edge and path zeta functions

11. Edge zeta functions
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. 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

The book ends with References and an Index.