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A family of Neumaier graphs containing examples with exactly five eigenvalues

Bart De Bruyn, Rhys J. Evans, Sergey Goryainov, Jack Koolen

Abstract

A Neumaier graph is an edge-regular graph with a regular clique. Such a graph is said to have parameters $(v,k,λ;e,s)$ if it is a $k$-regular graph on $v$ vertices having a clique of size $s$ such that every edge is contained in $λ$ 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 $(v,k,λ;e,s)=(48,14,2;1,4)$, among which 25 have exactly five eigenvalues. These 1063 graphs are also the first known examples of Neumaier graphs for the mentioned parameters.

A family of Neumaier graphs containing examples with exactly five eigenvalues

Abstract

A Neumaier graph is an edge-regular graph with a regular clique. Such a graph is said to have parameters if it is a -regular graph on vertices having a clique of size such that every edge is contained in triangles and every vertex outside is adjacent with exactly vertices inside . 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 , among which 25 have exactly five eigenvalues. These 1063 graphs are also the first known examples of Neumaier graphs for the mentioned parameters.
Paper Structure (5 sections, 3 theorems, 4 equations, 4 tables)

This paper contains 5 sections, 3 theorems, 4 equations, 4 tables.

Table of Contents

  1. Introduction
  2. A new family of Neumaier graphs
  3. Discussion of the cases $(\mu,\lambda) \in \{ (1,1),(1,3),(2,1) \}$
  4. Treatment of the case $(\lambda,\mu)=(2,2)$
  5. Determination of the spectrum for the graphs in families 1, 4 and 34

Key Result

Theorem 2.1

The graph $\overline{\Gamma}$ satisfies the following properties:

Theorems & Definitions (6)

  • Theorem 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 4.1
  • proof