On the number of fixed points of automorphisms of vertex-transitive graphs of bounded valency

The main result of this paper is that, if $\mathrm{\Gamma }$ is a finite connected $4$-valent arc-transitive graph, then either $\mathrm{\Gamma }$ is part of a well-understood family of graphs, or every non-identity automorphism of $\mathrm{\Gamma }$ fixes at most $1\mathrm{/}3$ of the vertices. As a corollary, we get a similar result for $3$-valent vertex-transitive graphs. Based on these results we propose a conjecture on the number of fixed points of non-identity automorphisms of vertex-transitive graphs of bounded valency.