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 invert-ible edges. Define 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 automor-phisms acting on a Riemann surface of genus g. In the present paper, we investigate cyclic group (Formula presented)_nacting purely harmonically on a graph X of genus g with fixed points. Given subgroup (Formula presented)_d< (Formula presented)_n, we find the signature of orbifold X/(Formula presented)_dthrough the signature of orbifold X/(Formula presented)_n. As a result, we obtain formulas for the number of fixed points for generators of group (Formula presented)_dand for genus of orbifold X/(Formula presented)_d. For Riemann surfaces, similar results were obtained earlier by M. J. Moore.