Ihara Zeta Functions Of Irregular Graphs
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Ihara Zeta Functions of Irregular Graphs
We explore three seemingly disparate but related avenues of inquiry: expanding what is known about the properties of the poles of the Ihara zeta function, determining what information about a graph is recoverable from its Ihara zeta function, and strengthening the ties between the Ihara zeta functions of graphs which are related to each other through common operations on graphs. Using the singular value decomposition of directed edge matrices, we give an alternate proof of the bounds on the poles of Ihara zeta functions. We then give an explicit formula for the inverse of directed edge matrices and use the inverse to demonstrate that the sum of the poles of an Ihara zeta function is zero. Next we discuss the information about a graph recoverable from its Ihara zeta function and prove that the girth of a graph as well as the number of cycles whose length is the girth can be read directly off of the reciprocal of the Ihara zeta function. We demonstrate that a graph's chromatic polynomial cannot in general be recovered from its Ihara zeta function and describe a method for constructing families of graphs which have the same chromatic polynomial but different Ihara zeta functions. We also show that a graph's Ihara zeta function cannot in general be recovered from its chromatic polynomial. Then we make the deletion of an edge from a graph less jarring (from the perspective of Ihara zeta functions) by viewing it as the limit as k goes to infinity of the operation of replacing the edge in the original graph we wish to delete with a walk of length k. We are able to prove that the limit of the Ihara zeta functions of the resulting graphs is in fact the Ihara zeta function of the original with the edge deleted. We also improve upon the bounds on the poles of the Ihara zeta function by considering digraphs whose adjacency matrices are directed edge matrices.
Zeta Functions of Graphs
Author: Audrey Terras
language: en
Publisher: Cambridge University Press
Release Date: 2010-11-18
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.
Analysis on Graphs and Its Applications
This book addresses a new interdisciplinary area emerging on the border between various areas of mathematics, physics, chemistry, nanotechnology, and computer science. The focus here is on problems and techniques related to graphs, quantum graphs, and fractals that parallel those from differential equations, differential geometry, or geometric analysis. Also included are such diverse topics as number theory, geometric group theory, waveguide theory, quantum chaos, quantum wiresystems, carbon nano-structures, metal-insulator transition, computer vision, and communication networks.This volume contains a unique collection of expert reviews on the main directions in analysis on graphs (e.g., on discrete geometric analysis, zeta-functions on graphs, recently emerging connections between the geometric group theory and fractals, quantum graphs, quantum chaos on graphs, modeling waveguide systems and modeling quantum graph systems with waveguides, control theory on graphs), as well as research articles.