Vertex-transitive Neumaier graphs

TL;DR

This paper investigates vertex-transitive Neumaier graphs, providing conditions for when they are strongly regular and identifying certain Cayley graphs with small valency.

Contribution

It establishes a necessary and sufficient condition for vertex-transitive Neumaier graphs to be strongly regular and classifies small valency Cayley graphs within this class.

Findings

Characterization of when vertex-transitive Neumaier graphs are strongly regular.

Identification of Neumaier Cayley graphs with valency at most 10.

Insights into the structure of vertex-transitive Neumaier graphs.

Abstract

A graph $\Gamma$ is called edge-regular whenever it is regular and for any two adjacent vertices, the number of their common neighbors is independent of the choice of vertices. A clique $C$ in $\Gamma$ is called regular whenever for any vertex out of $C$, the number of its neighbors in $C$ is independent of the vertex. A Neumaier graph is a non-complete edge-regular graph with a regular clique. In this paper, we study vertex-transitive Neumaier graphs. We give a necessary and sufficient condition under which a vertex-transitive Neumaier graph is strongly regular. We also identify Neumaier Cayley graphs with small valency at most $10$ among vertex-transitive Neumaier graphs.