Fixed points of cyclic groups acting purely harmonically on a graph

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.