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arXiv · 2002.07275

Zeta functions of edge-free quotients of graphs

Dmitry Zakharov
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Abstract

We consider the Ihara zeta function $\zeta(u,X//G)$ and Artin-Ihara $L$-function of the quotient graph of groups $X//G$, where $G$ is a group acting on a finite graph $X$ with trivial edge stabilizers. We determine the relationship between the primes of $X$ and $X//G$ and show that $X\to X//G$ can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the $L$-function of $X//G$ evaluated at the regular representation is equal to $\zeta(u,X)$ and that $\zeta(u,X//G)$ divides $\zeta(u,X)$. We derive two-term and three-term determinant formulas for the zeta and $L$-functions, and compute several examples of $L$-functions of edge-free quotients of the tetrahedron graph $K_4$.

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Dmitry Zakharov. 2020-02-17. Zeta functions of edge-free quotients of graphs. https://arxiv.org/abs/2002.07275

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