An infinite class of Neumaier graphs and non-existence results

TL;DR

This paper introduces a new construction method for strictly Neumaier graphs, demonstrating infinitely many instances, and establishes necessary conditions that exclude certain parameter sets from existing.

Contribution

It provides a novel construction technique for strictly Neumaier graphs and proves their infinite existence using Jacobi sums, along with non-existence results for specific parameters.

Findings

Infinitely many strictly Neumaier graphs constructed

Necessary conditions for Neumaier graph existence established

Certain parameter sets proven non-admissible

Abstract

A Neumaier graph is a non-complete edge-regular graph containing a regular clique. A Neumaier graph that is not strongly regular is called a strictly Neumaier graph. In this work we present a new construction of strictly Neumaier graphs, and using Jacobi sums, we show that our construction produces infinitely many instances. Moreover, we prove some necessary conditions for the existence of (strictly) Neumaier graphs that allow us to show that several parameter sets are not admissible.