Zeta Functions of Graphs: A Stroll through the Garden

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Cambridge University Press, Nov 18, 2010 - Mathematics
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Graph 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.
 

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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
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About the author (2010)

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

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