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 P G(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.
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 P G(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.
| Original language | English |
|---|---|
| Publisher | arXiv |
| Publication status | Published - Apr 2024 |
Keywords
- divisible design graph
- symplectic graphs
- spread
- equitable partition
- projective space