On fixity of arc-transitive graphs

TL;DR

This paper proves that the relative fixity of large connected 2-arc-transitive graphs with fixed valence approaches zero as the number of vertices increases, extending to various classes of arc-transitive graphs.

Contribution

It establishes that the relative fixity tends to zero for large graphs in specific arc-transitive classes, generalizing previous results to broader graph classes.

Findings

Relative fixity of 2-arc-transitive graphs tends to 0 as vertices grow.

Same result holds for fixed prime valence arc-transitive graphs.

Generalizes to locally-L graphs with L quasiprimitive and graph-restrictive.

Abstract

The relative fixity of a permutation group is the maximum proportion of the points fixed by a non-trivial element of the group and the relative fixity of a graph is the relative fixity of its automorphism group, viewed as a permutation group on the vertex-set of the graph. We prove in this paper that the relative fixity of connected $2$-arc-transitive graphs of a fixed valence tends to $0$ as the number of vertices grows to infinity. We prove the same result for the class of arc-transitive graphs of a fixed prime valence, and more generally, for any class of arc-transitive locally-$L$ graphs, where $L$ is a fixed quasiprimitive graph-restrictive permutation group.