Divisible design graphs

Abstract

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of ( v , k , λ ) -graphs, and like ( v , k , λ ) -graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the eigenvalues of the adjacency matrix, we obtain necessary conditions for existence. Old results of Bose and Connor on symmetric divisible designs give other conditions and information on the structure. Many constructions are given using various combinatorial structures, such as ( v , k , λ ) -graphs, distance-regular graphs, symmetric divisible designs, Hadamard matrices, and symmetric balanced generalized weighing matrices. Several divisible design graphs are characterized in terms of the parameters.