Characteristic polynomials of digraphs having a semi-free action

Let ϕ be a semi-free action of a group G on a finite digraph Γ. The front divisor Γ/π(ϕ) of Γ has as vertex set the set of the vertex orbits of ϕ and there are t arcs going from u¯ to v¯ in Γ/π(ϕ) if from any vertex u of the orbit u¯ there are t arcs of Γ going towards the vertices in v¯. Our main result is that the characteristic polynomial of Γ is a product of the characteristic polynomial of its front divisor Γ/π(ϕ) and polynomials associated with the free part of Γ under ϕ. This work extends earlier work of Lee and Kim [Characteristic polynomials of graphs having a semifree action, Linear Algebra Appl. 307 (2000) 35-46].