New versions of the Wallis-Fon-Der-Flaass construction to create divisible design graphs

TL;DR

This paper modifies the Wallis-Fon-Der-Flaass construction to generate new divisible design graphs, expanding the toolkit for constructing such graphs and sometimes producing strongly regular graphs.

Contribution

It introduces new modifications to an existing construction method, enabling the creation of a broader class of divisible design graphs and some strongly regular graphs.

Findings

New constructions of divisible design graphs are presented.

Some constructions result in strongly regular graphs.

The modifications extend the applicability of the original Wallis-Fon-Der-Flaass method.

Abstract

A k-regular graph on v vertices is a divisible design graph with parameters (v, k, lambda_1 ,lambda_2, m, n) if its vertex set can be partitioned into m classes of size n, such that 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 whenever it is not complete or edgeless. If m=1, then a divisible design graph is strongly regular with parameters (v, k, lambda_1, lambda_1). In this paper the Wallis-Fon-Der-Flaass construction of strongly regular graphs is modified to create new constructions of divisible design graphs. In some cases, these constructions lead to strongly regular graphs.