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Zeta functions of graphs with $\mathbb{Z}$ actions

Bryan Clair
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Abstract

Suppose $Y$ is a regular covering of a graph $X$ with covering transformation group $π= \mathbb{Z}$. This paper gives an explicit formula for the $L^2$ zeta function of $Y$ and computes examples. When $π= \mathbb{Z}$, the $L^2$ zeta function is an algebraic function. As a consequence it extends to a meromorphic function on a Riemann surface. The meromorphic extension provides a setting to generalize known properties of zeta functions of regular graphs, such as the location of singularities and the functional equation.

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Bryan Clair. 2006-07-26. Zeta functions of graphs with $\mathbb{Z}$ actions. https://arxiv.org/abs/math/0607689

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