Divisible design graphs from Higmanian association schemes

TL;DR

This paper explores the connection between Higmanian association schemes and divisible design graphs, providing conditions for their relationship, unifying known examples, and constructing new infinite families.

Contribution

It establishes conditions linking Higmanian association schemes to divisible design graphs and introduces new infinite families through fusion techniques.

Findings

Several known divisible design graphs are shown to be fusions of Higmanian schemes

Conditions are provided for unions of relations to form divisible design graphs

New infinite families of divisible design graphs are constructed

Abstract

An imprimitive symmetric indecomposable association scheme of rank 5 is said to be Higmanian. A divisible design graph is a graph whose adjacency matrix is an incidence matrix of a symmetric divisible design. We establish conditions which guarantee that a union of some basis relations of a Higmanian association scheme is an edge set of a divisible design graph. Further, we show that several known families of divisible design graphs can be obtained as fusions of Higmanian association schemes. Finally, using our approach we construct new infinite families of divisible design graphs.