Fixed points of cyclic groups acting purely harmonically on a graph

Abstract

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. Define a genus g of the graph X to be the rank of the first homology group. A discrete version of the Wiman theorem states that the order of a cyclic group $\mathbb{Z}_n$ 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 fixed points has an automorphism generating a <<large>> cyclic group $\mathbb{Z}_n$ of order $n\ge2g-1.$ We show that in the most cases, the automorphism acts fixed point free, while for groups of order 2g and 2g-1 it can have one or two fixed points.