On the Spectrum of a Weakly Distance-Regular Digraph

Abstract

The notion of distance-regularity for undirected graphs can be extended for the directed case in two dierent ways. Damerell adopted the strongest definition of distanceregularity, which is equivalent to say that the corresponding set of distance matrices {Ai} D=0 constitutes a commutative association scheme. In particular, a (strongly) distance-regular digraph is stable, which means that A > = Ag i, for each i = 1,...,g 1, where g denotes the girth of . If we remove the stability property from the definition of distance-regularity, it still holds that the number of walks of a given length between any two vertices of does not depend on the chosen vertices but only on their distance. We consider the class of digraphs characterized by such a weaker condition, referred to as weakly distance-regular digraphs, and show that their spectrum can also be obtained from a smaller ‘quotient digraph’. As happens in the case of distance-regular graphs, the study is greatly facilitated by a family of orthogonal polynomials called the distance polynomials.