Characteristic polynomials of digraphs having a semi-free action

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