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Divisible design graphs from the symplectic graph
De Bruyn,Bart Goryainov,Sergey Haemers,Willem H. Shalaginov,Leonid
De Bruyn,Bart
Goryainov,Sergey
Haemers,Willem H.
Shalaginov,Leonid
Abstract
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≥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 that consists of lines which are not totally isotropic and which is closed under the action of the associated symplectic polarity. Existence of symplectic spreads is known, but the construction of a special spread for every odd prime power q is a main result of this paper. We also show an equivalence between special spreads of Sp(4, q) and certain nice point sets in the projective space PG(4,q). We have included relevant background from finite geometry, and when q=3,5 and 7 we worked out all possible special spreads.
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Publisher Copyright: © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2024.
Date
2025-05
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s10623-024-01557-w.pdf
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Keywords
Divisible design graph, Equitable partition, Projective space, Spread, Symplectic graph
Citation
De Bruyn, B, Goryainov, S, Haemers, W H & Shalaginov, L 2025, 'Divisible design graphs from the symplectic graph', Designs Codes and Cryptography, vol. 93, no. 5, pp. 1401-1424. https://doi.org/10.1007/s10623-024-01557-w