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

TL;DR

This paper proves that in finite connected 4-valent arc-transitive graphs, non-identity automorphisms fix at most one-third of vertices unless the graph belongs to a known family, and extends similar results to 3-valent graphs, proposing a broader conjecture.

Contribution

It establishes bounds on fixed points of automorphisms in bounded valency vertex-transitive graphs and introduces a new conjecture based on these findings.

Findings

Non-identity automorphisms fix at most 1/3 of vertices in 4-valent arc-transitive graphs.

A similar fixed point bound applies to 3-valent vertex-transitive graphs.

Proposes a conjecture on fixed points for all bounded valency vertex-transitive graphs.

Abstract

The main result of this paper is that, if $\Gamma$ is a finite connected $4$-valent arc-transitive graph, then either $\Gamma$ is part of a well-understood family of graphs, or every non-identity automorphism of $\Gamma$ fixes at most $1/3$ 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.