Functional Equations and Pole Structure of the Bartholdi Zeta Function

TL;DR

This paper explores the properties of the Bartholdi zeta function on graphs, deriving functional equations, pole bounds, and conditions for the Riemann hypothesis, revealing deep connections with graph regularity and spectral properties.

Contribution

It introduces new functional equations and pole structure analyses for the Bartholdi zeta function on regular and general graphs, linking it to the Ihara zeta function and spectral conditions.

Findings

Derived a functional equation for the Bartholdi zeta function.

Identified pole bounds and their saturation conditions.

Established conditions under which the Riemann hypothesis holds for the zeta function.

Abstract

In this paper, we investigate the Bartholdi zeta function on a connected simple digraph with $n_V$ vertices and $n_E$ edges. We derive a functional equation for the Bartholdi zeta function $\zeta_G(q,u)$ on a regular graph $G$ with respect to the bump parameter $u$. We also find an equivalence between the Bartholdi zeta function with a specific value of $u$ and the Ihara zeta function at $u=0$. We determine bounds of the critical strip of $\zeta_G(q,u)$ for a general graph. If $G$ is a $(t+1)$-regular graph, the bounds are saturated and $q=(1-u)^{-1}$ and $q=(t+u)^{-1}$ are the poles at the boundaries of the critical strip for $u\ne 1, -t$. When $G$ is the regular graph and the spectrum of the adjacency matrix satisfies a certain condition, $\zeta_G(q,u)$ satisfies the so-called Riemann hypothesis. For $u \ne 1$, $q=\pm(1-u)^{-1}$ are poles of $\zeta_G(q,u)$ unless $G$ is tree. Although the order of the pole at $q=(1-u)^{-1}$ is $n_E-n_V+1$ if $u\ne u_* \equiv 1-\frac{n_E}{n_V}$, it is enhanced at $u=u_*$. In particular, if the Moore-Penrose inverse of the incidence matrix $L^+$ and the degree vector $\vec{d}$ satisfy the condition $|L^+ \vec{d}|^2\ne n_E$, the order of the pole at $q=(1-u)^{-1}$ increases only by one at $u=u_*$. The order of the pole at $q=-(1-u)^{-1}$ coincides with that at $q=(1-u)^{-1}$ if $G$ is bipartite and is $n_E-n_V$ otherwise.