## Zeta Functions of Graphs: A Stroll through the GardenGraph 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. |

### What people are saying - Write a review

We haven't found any reviews in the usual places.

### Contents

II Ihara zeta function and the graph theory prime number theorem | 43 |

III Edge and path zeta functions | 81 |

IV Finite unramified Galois coverings of connected graphs | 103 |

V Last look at the garden | 209 |

### Other editions - View all

### Common terms and phrases

adjacency matrix algebraic number analog Artin L-functions base graph bipartite Cayley graph circle has radius conjecture conjugacy class corresponding cosets covering graphs cube cycle Dedekind zeta function deﬁned Deﬁnition determinant formula directed edge edge adjacency matrix edge L-function edge zeta function eigenvalues elements entries equivalent example Exercise Figure ﬁnd ﬁrst ﬁxed follows Frobenius automorphism fundamental group Galois group Galois theory graph coverings graph theory group G histogram Hoory Ihara zeta function inﬁnite intermediate cover intermediate graphs intermediate to Y/X irreducible isomorphic isospectral k-regular Lemma length level spacings lift loops Lubotzky Math normal cover normalized Frobenius automorphism number ﬁelds number of vertices number theory permutation prime number theorem proof Proposition prove quadratic cover Ramanujan graphs random regular graphs Riemann hypothesis Sarnak satisﬁes Selberg sheet Siegel pole spanning tree spectrum subgroup of G Terras tetrahedron usual hypotheses vector zeros