Divisible design graphs from the symplectic graph
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866913315646603264 |
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| author | De Bruyn, Bart Goryainov, Sergey Haemers, Willem Shalaginov, Leonid |
| author_facet | De Bruyn, Bart Goryainov, Sergey Haemers, Willem Shalaginov, Leonid |
| contents | A divisible design graph is a graph whose adjacency matrix is an incidence matrix of a (group) divisible design. Divisible design graphs were introduced in 2011 as a generalization of $(v,k,λ)$-graphs. Here we describe four new infinite families that can be obtained from the symplectic strongly regular graph $Sp(2e,q)$ ($q$ odd, $e\geq 2$) by modifying the set of edges. To achieve this we need two kinds of spreads in $PG(2e-1,q)$ with respect to the associated symplectic form: the symplectic spread consisting of totally isotropic subspaces and, when $e=2$, a special spread consisting of lines which are not totally isotropic. Existence of symplectic spreads is known, but the construction of a special spread for every odd prime power $q$ is a major result of this paper. We have included relevant back ground from finite geometry, and when $q=3,5$ and $7$ we worked out all possible special spreads. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_09902 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Divisible design graphs from the symplectic graph De Bruyn, Bart Goryainov, Sergey Haemers, Willem Shalaginov, Leonid Combinatorics 05B25, 05E30, 51A50 A divisible design graph is a graph whose adjacency matrix is an incidence matrix of a (group) divisible design. Divisible design graphs were introduced in 2011 as a generalization of $(v,k,λ)$-graphs. Here we describe four new infinite families that can be obtained from the symplectic strongly regular graph $Sp(2e,q)$ ($q$ odd, $e\geq 2$) by modifying the set of edges. To achieve this we need two kinds of spreads in $PG(2e-1,q)$ with respect to the associated symplectic form: the symplectic spread consisting of totally isotropic subspaces and, when $e=2$, a special spread consisting of lines which are not totally isotropic. Existence of symplectic spreads is known, but the construction of a special spread for every odd prime power $q$ is a major result of this paper. We have included relevant back ground from finite geometry, and when $q=3,5$ and $7$ we worked out all possible special spreads. |
| title | Divisible design graphs from the symplectic graph |
| topic | Combinatorics 05B25, 05E30, 51A50 |
| url | https://arxiv.org/abs/2404.09902 |