Two Moore's theorems for graphs

Let X be a finite connected graph, possibly with loops and multiple edges. An automorphism group of X acts purely harmonically if it acts freely on the set of directed edges of X and has no invertible edges. Dene a genus g of the graph X to be the rank of the first homology group. A finite group acting purely harmonically on a graph of genus g is a natural discrete analogue of a finite group of automorphisms acting on a Riemann surface of genus g: In the present paper, we investigate cyclic group Zn acting purely harmonically on a graph X of genus g with fixed points. Given subgroup Zd n; we find the signature of orbifold X=Zd through the signature of orbifold X=Zn: As a result, we obtain formulas for the number of fixed points for generators of group Zd and for genus of orbifold X=Zd: For Riemann surfaces, similar results were obtained earlier by M. J. Moore