Vertex-primitive digraphs with large fixity

TL;DR

This paper characterizes vertex-primitive digraphs with high relative fixity, showing finiteness results for those with bounded out-valency and fixity above a positive threshold.

Contribution

It provides a complete characterization of vertex-primitive digraphs with relative fixity at least 1/3 and establishes finiteness for those with bounded out-valency and high fixity.

Findings

Vertex-primitive digraphs with relative fixity ≥ 1/3 are characterized.

Finitely many such digraphs exist with bounded out-valency and fixity above a positive constant.

Abstract

The relative fixity of a digraph $\Gamma$ is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of $\Gamma$ and the number of vertices of $\Gamma$. We characterize the vertex-primitive digraphs whose relative fixity is at least $1/3$, and we show that there are only finitely many vertex-primitive digraphs of bounded out-valency and relative fixity exceeding a positive constant.