Zeta functions of edge-free quotients of graphs

We define the Ihara zeta function ζ(u,X) and Artin–Ihara L-function of quotient graph groups X=Y//G, where G is a group acting on finite Y with trivial edge stabilizers. determine relationship between primes X show that projection map Y→X can be naturally viewed as an unramified Galois covering graphs groups. evaluated at regular representation equal to ζ(u,Y), divides ζ(u,Y). derive two-term three-term determinant formulas for L-functions, compute several examples L-functions edge-free quotients tetrahedron K4.