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arXiv · 2603.17029

A family of Neumaier graphs containing examples with exactly five eigenvalues

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

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Bart De Bruyn, Rhys J. Evans, Sergey Goryainov, Jack Koolen. 2026-03-17. A family of Neumaier graphs containing examples with exactly five eigenvalues. https://arxiv.org/abs/2603.17029

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