Construction of divisible design graphs using affine designs
Vladislav V. Kabanov
TL;DR
This work addresses constructing divisible design graphs by developing a new construction that blends a symmetric $2$-design with affine designs and a network of bijections between parallel classes to control interclass adjacency; this yields $v = q^d m$, $k = q^{d-1}(q^d - 1)$, $\\lambda_1 = q^{d-1}(q^d - q^{d-1} - 1)$ and $\\lambda_2 = q^{d-2}(q-1)^2 \\\lambda$, with $m$, $n$ tied to a chosen $q$ and $d$; the paper also introduces a partial complement variant yielding updated parameters and shows how such graphs relate to symplectic graphs over rings, including explicit connections to $X(2e,K)$ and $Y(2e,K)$ and computational verifications; overall, it expands the construction toolkit for divisible design graphs and links classical design theory with ring-theoretic and affine-geometric constructions.
Abstract
A $k$-regular graph on $v$ vertices is a {\em divisible design graph} if there exist integers $λ_1,λ_2,m,n$ such that the vertex set can be partitioned into $m$ classes of size $n$ and any two different vertices from the same class have $λ_1$ common neighbours, and any two vertices from different classes have $λ_2$ common neighbours. In this paper, a new construction that produces divisible design graphs is provided.