On the existence of small strictly Neumaier graphs
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914700381388800 |
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| author | Abiad, Aida De Boeck, Maarten Zeijlemaker, Sjanne |
| author_facet | Abiad, Aida De Boeck, Maarten Zeijlemaker, Sjanne |
| contents | A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this work, we prove several results on the existence of small strictly Neumaier graphs. In particular, we present a theoretical proof of the uniqueness of the smallest strictly Neumaier graph with parameters $(16,9,4;2,4)$, we establish the existence of a strictly Neumaier graph with parameters $(25,12,5;2,5)$, and we disprove the existence of strictly Neumaier graphs with parameters $(25,16,9;3,5)$, $(28,18,11;4,7)$, $(33,24,17;6,9)$, $(35,22,12;3,5)$ and $(55,34,18;3,5)$. Our proofs use combinatorial techniques and a novel application of integer programming methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_15406 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the existence of small strictly Neumaier graphs Abiad, Aida De Boeck, Maarten Zeijlemaker, Sjanne Combinatorics A Neumaier graph is a non-complete edge-regular graph containing a regular clique. In this work, we prove several results on the existence of small strictly Neumaier graphs. In particular, we present a theoretical proof of the uniqueness of the smallest strictly Neumaier graph with parameters $(16,9,4;2,4)$, we establish the existence of a strictly Neumaier graph with parameters $(25,12,5;2,5)$, and we disprove the existence of strictly Neumaier graphs with parameters $(25,16,9;3,5)$, $(28,18,11;4,7)$, $(33,24,17;6,9)$, $(35,22,12;3,5)$ and $(55,34,18;3,5)$. Our proofs use combinatorial techniques and a novel application of integer programming methods. |
| title | On the existence of small strictly Neumaier graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2308.15406 |