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## Zeta Functions of Graphs: A Stroll through the Garden (Hardback)"Graph theory meets number theory in this stimulating book." — From the back cover ## $43.95 |

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Please contact us if you've got questions about this product.## 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.

## Description

This is #128 in the *Cambridge 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**

- Riemann zeta functions and other zetas from number theory
- Ihara zeta functions
- Selberg zeta functions
- Ruelle zeta functions
- Chaos

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

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

**Part III. Edge and path zeta functions**

- Edge zeta functions
- Path zeta functions

**Part IV. Finite unramified Galois coverings of connected graphs**

- Finite unramified coverings and Galois groups
- Fundamental theorem of Galois theory
- Behavior of primes in coverings
- Frobenius automorphisms
- How to construct intermediate coverings using the Frobenius automorphism
- Artin
*L*-functions - Edge Artin
*L*-functions - Path Artin
*L*-functions - Non-isomorphic regular graphs without loops or multiedges having the same Ihara zeta function
- Chebotarev density theorem
- Siegel poles

**Part V. Last look at the garden**

- An application to error-correcting codes
- Explicit formulas
- Again chaos
- Final research problems

The book ends with References and an Index.

## Details

- SKU:
- 2222
- Subject:
- Mathematics
- Author:
- Terras, Audrey
- Publisher:
- Cambridge University Press
- Publish Date:
- 2011
- Format:
- Hardback
- Weight (pounds):
- 1.16
- Dimensions (W”xL”xH”):
- 6.2″x9.25″x0.8″
- ISBN-13:
- 9780521113670
- ISBN-10:
- 0521113679
- Grade Level:
- Graduate
- Series:
- Cambridge Studies in Advanced Mathematics

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