Zeta functions of edge-free quotients of graphs

We consider the Ihara zeta function $\zeta (u,X\mathrm{/}\mathrm{/}G)$ and Artin-Ihara $L$-function of the quotient graph of groups $X\mathrm{/}\mathrm{/}G$, 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 $X\mathrm{/}\mathrm{/}G$ and show that $X\to X\mathrm{/}\mathrm{/}G$ can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the $L$-function of $X\mathrm{/}\mathrm{/}G$ evaluated at the regular representation is equal to $\zeta (u,X)$ and that $\zeta (u,X\mathrm{/}\mathrm{/}G)$ 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 $K_4$.