We define the Ihara zeta function ζ(u,X) and Artin–Ihara L-function of the quotient graph of groups 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 X and show that the projection map Y→X can be naturally viewed as an unramified Galois covering of graphs of groups. We show that the L-function of X evaluated at the regular representation is equal to ζ(u,Y), and that ζ(u,X) divides ζ(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 K4.
- Graph of groups
- Ihara zeta function