Fixed points of cyclic groups acting purely harmonically on a graph

Abstract

Let X be a nite 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 rst homology group. A discrete version of the Wiman theorem states that the order of a cyclic group Zn acting purely harmonically on a graph X of genus g > 1 is bounded from above by 2g + 2. In the present paper, we investigate how many xed points has an automorphism generating a ¾large¿ cyclic group Zn of order n ≥ 2g − 1. We show that in the most cases, the automorphism acts xed point free, while for groups of order 2g and 2g − 1 it can have one or two xed points.