Functional Equations and Pole Structure of the Bartholdi Zeta Function

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.