Characteristic polynomials of digraphs having a semi-free action

Let phi be a semi-free action of a group G on a finite digraph Gamma. The front divisor Gamma/pi(phi) of Gamma has as vertex set the set of the vertex orbits of phi and there are t arcs going from it to (v) over bar in Gamma/pi(phi) if from any vertex (u) over bar of the orbit (u) over bar there are t arcs of Gamma going towards the vertices in (v) over bar. Our main result is that the characteristic polynomial of Gamma is a product of the characteristic polynomial of its front divisor Gamma/pi(phi) and polynomials associated with the free part of Gamma under phi. This work extends earlier work of Lee and Kim.