Divisible design graphs from Higmanian association schemes

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.