We define the Ihara zeta function $\zeta (u,\mathbb{X})$ and Artin–Ihara L-function of the quotient graph of groups $\mathbb{X}=Y//G$, where G is a group acting on a finite graph Y with trivial edge stabilizers. We determine the relationship between the primes of Y and $\mathbb{X}$ and show that the projection map $Y\to \mathbb{X}$ can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the L-function of $\mathbb{X}$ evaluated at the regular representation is equal to $\zeta (u,Y)$, and that $\zeta (u,\mathbb{X})$ divides $\zeta (u,Y)$. 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 $K_4$.

我们定义 Ihara zeta 函数 $\zeta (你,\mathbb{X})$和 Artin-Ihara L - 群商图的函数$\mathbb{X}=是//G$，其中G是作用在有限图Y上的群，带有平凡的边稳定器。我们确定Y的素数和$\mathbb{X}$ 并显示投影图 $是\to \mathbb{X}$可以很自然地将其视为群图的无分支伽罗瓦覆盖。我们证明了L函数$\mathbb{X}$ 以正则表示求值等于 $\zeta (你,是)$， 然后 $\zeta (你,\mathbb{X})$ 分裂 $\zeta (你,是)$. 我们推导了zeta和L函数的二项和三项行列式公式，并计算了四面体图的无边商的L函数的几个例子${钾}_4$.