Uniformly vertex-transitive graphs

TL;DR

This paper introduces uniformly vertex-transitive graphs, explores their properties using derangement graphs, and classifies certain small examples, addressing open questions in graph symmetry.

Contribution

It defines uniformly vertex-transitive graphs, establishes their characterization via derangement graphs, and classifies small non-Cayley examples.

Findings

Existence of vertex-transitive graphs that are not uniformly vertex-transitive.

Criteria for uniform vertex-transitivity with imprimitive automorphism groups.

Complete classification of non-Cayley uniformly vertex-transitive graphs under 30 vertices.

Abstract

We introduce uniformly vertex-transitive graphs as vertex-transitive graphs satisfying a stronger condition on their automorphism groups, motivated by a problem which arises from a Sinkhorn-type algorithm. We use the derangement graph $D(\Gamma)$ of a given graph $\Gamma$ to show that the uniform vertex-transitivity of $\Gamma$ is equivalent to the existence of cliques of sufficient size in $D(\Gamma)$. Using this method, we find examples of graphs that are vertex-transitive but not uniformly vertex-transitive, settling a previously open question. Furthermore, we develop sufficient criteria for uniform vertex-transitivity in the situation of a graph with an imprimitive automorphism group. We classify the non-Cayley uniformly vertex-transitive graphs on less than 30 vertices outside of two complementary pairs of graphs.