Uniformly vertex-transitive graphs
Simon Schmidt, Chase Vogeli, Moritz Weber

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 of a given graph to show that the uniform vertex-transitivity of is equivalent to the existence of cliques of sufficient size in . 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.
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Taxonomy
TopicsFinite Group Theory Research · Advanced Graph Theory Research · Advanced Operator Algebra Research
