A family of Neumaier graphs containing examples with exactly five eigenvalues
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910057504964608 |
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| author | De Bruyn, Bart Evans, Rhys J. Goryainov, Sergey Koolen, Jack |
| author_facet | De Bruyn, Bart Evans, Rhys J. Goryainov, Sergey Koolen, Jack |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17029 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A family of Neumaier graphs containing examples with exactly five eigenvalues De Bruyn, Bart Evans, Rhys J. Goryainov, Sergey Koolen, Jack Combinatorics 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. |
| title | A family of Neumaier graphs containing examples with exactly five eigenvalues |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2603.17029 |