A family of Neumaier graphs containing examples with exactly five eigenvalues

TL;DR

This paper introduces a new family of Neumaier graphs, including 25 with exactly five eigenvalues, addressing an open problem about their existence and providing the first known examples for specific parameters.

Contribution

The paper constructs a family of Neumaier graphs and identifies 1063 nonisomorphic graphs with specific parameters, including 25 with exactly five eigenvalues, solving an open problem.

Findings

Found 1063 nonisomorphic Neumaier graphs with parameters (48,14,2;1,4)

Identified 25 graphs within this family that have exactly five eigenvalues

Provided the first known examples of Neumaier graphs with these parameters

Abstract

A Neumaier graph is an edge-regular graph with a regular clique. Such a graph is said to have parameters $(v,k,\lambda;e,s)$ if it is a $k$-regular graph on $v$ vertices having a clique of size $s$ such that every edge is contained in $\lambda$ triangles and every vertex outside $C$ is adjacent with exactly $e$ vertices inside $C$. It was an open problem whether Neumaier graphs can exist with exactly five eigenvalues. In the present paper, we describe a family of Neumaier graphs, and show that inside this family there are 1063 nonisomorphic Neumaier graphs with parameters $(v,k,\lambda;e,s)=(48,14,2;1,4)$, among which 25 have exactly five eigenvalues. These 1063 graphs are also the first known examples of Neumaier graphs for the mentioned parameters.