Construction of divisible design graphs using affine designs

TL;DR

This paper introduces a new method for constructing divisible design graphs, which are special regular graphs with a partitioned vertex set and specific common neighbor properties.

Contribution

The paper presents a novel construction technique for divisible design graphs based on affine designs, expanding the known classes of such graphs.

Findings

New construction method for divisible design graphs

Extension of affine design applications in graph theory

Potential for generating diverse divisible design graphs

Abstract

A $k$-regular graph on $v$ vertices is a {\em divisible design graph} if there exist integers $\lambda_1,\lambda_2,m,n$ such that the vertex set can be partitioned into $m$ classes of size $n$ and any two different vertices from the same class have $\lambda_1$ common neighbours, and any two vertices from different classes have $\lambda_2$ common neighbours. In this paper, a new construction that produces divisible design graphs is provided.