The smallest strictly Neumaier graph and its generalisations

TL;DR

This paper characterizes the smallest strictly Neumaier graph with 16 vertices, explores parameter tuples for such graphs up to 24 vertices, and constructs infinite sequences of these graphs with specific properties, addressing recent open questions.

Contribution

It determines the smallest strictly Neumaier graph, classifies all such graphs up to 24 vertices, and constructs infinite families with regular cliques, advancing understanding of Neumaier graphs.

Findings

Smallest strictly Neumaier graph has 16 vertices.

All strictly Neumaier graphs with up to 24 vertices are characterized.

Constructed infinite sequences of strictly Neumaier graphs with regular cliques.

Abstract

A regular clique in a regular graph is a clique such that every vertex outside of the clique is adjacent to the same positive number of vertices inside the clique. We continue the study of regular cliques in edge-regular graphs initiated by A. Neumaier in the 1980s and attracting current interest. We thus define a Neumaier graph to be an non-complete edge-regular graph containing a regular clique, and a strictly Neumaier graph to be a non-strongly regular Neumaier graph. We first prove some general results on Neumaier graphs and their feasible parameter tuples. We then apply these results to determine the smallest strictly Neumaier graph, which has $16$ vertices. Next we find the parameter tuples for all strictly Neumaier graphs having at most $24$ vertices. Finally, we give two sequences of graphs, each with $i^{\text{th}}$ element a strictly Neumaier graph containing a $2^{i}$-regular clique (where $i$ is a positive integer) and having parameters of an affine polar graph as an edge-regular graph. This answers questions recently posed by G. Greaves and J. Koolen.