On the existence of small strictly Neumaier graphs

TL;DR

This paper investigates the existence of small strictly Neumaier graphs, proving uniqueness for one set of parameters, establishing existence for another, and disproving several others using combinatorial and integer programming techniques.

Contribution

It provides new theoretical results on the existence and uniqueness of small strictly Neumaier graphs with specific parameters.

Findings

Proved the uniqueness of the smallest strictly Neumaier graph with parameters (16,9,4;2,4)

Established the existence of a strictly Neumaier graph with parameters (25,12,5;2,5)

Disproved the existence of strictly Neumaier graphs with several other parameter sets

Abstract

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this work, we prove several results on the existence of small strictly Neumaier graphs. In particular, we present a theoretical proof of the uniqueness of the smallest strictly Neumaier graph with parameters $(16,9,4;2,4)$, we establish the existence of a strictly Neumaier graph with parameters $(25,12,5;2,5)$, and we disprove the existence of strictly Neumaier graphs with parameters $(25,16,9;3,5)$, $(28,18,11;4,7)$, $(33,24,17;6,9)$, $(35,22,12;3,5)$ and $(55,34,18;3,5)$. Our proofs use combinatorial techniques and a novel application of integer programming methods.