Alternating Walk/Zeta Correspondence

TL;DR

This paper introduces a new class of alternating zeta functions for graphs, expressing them via Ihara zeta functions, spectra, and Laplacians, and explores their limits and relations to Mahler measures.

Contribution

It defines a generalized alternating zeta function for graphs, relates it to spectral properties, and provides integral formulas and examples, advancing the understanding of graph zeta functions.

Findings

Expressed alternating zeta functions using Ihara zeta functions.

Derived spectral representations for vertex-transitive regular graphs.

Presented integral limits and relations to Mahler measures.

Abstract

We consider the alternating zeta function and the alternating $L$-function of a graph $G$, and express them by using the Ihara zeta function of $G$. Next, we define a generalized alternating zeta function of a graph, and express the generalized alternating zeta function of a vertex-transitive regular graph by spectra of the transition probability matrix of the symmetric simple random walk on it and its Laplacian. Furthermore, we present an integral expression for the limit of the generalized alternating zeta functions of a series of vertex-transitive regular graphs. As an example, we treat the generalized alternating zeta functions of a finite torus. Finally, we treat the relation between the Mahler measure and the alternating zeta function of a graph.