On fixity of arc-transitive graphs
Florian Lehner, Primoz Potocnik, Pablo Spiga

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 -arc-transitive graphs of a fixed valence tends to 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- graphs, where is a fixed quasiprimitive graph-restrictive permutation group.
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