[PDF] Zeta Functions of Graphs: A Stroll through the Garden (Cambridge Studies in Advanced Mathematics)

ISBN: 0521113679

Category: Tutorial


Posted on 2018-01-21, by luongquocchinh.

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Author: Audrey Terras | Publisher: Cambridge University Press | Category: Mathematics | Language: English | Page: 252 | ISBN: 0521113679 | ISBN13: 9780521113670 |

Description: 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. Pitched at beginning graduate students, the book will also appeal to researchers. Many well-chosen illustrations and diagrams, and exercises throughout, theoretical and computer-based.

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