On the Spectrum of a Weakly Distance-Regular Digraph

The notion of distance-regularity for undirected graphs can be extended for the directed case in two different ways. Damerell adopted the strongest definition of distance-regularity, which is equivalent to say that the corresponding set of distance matrices {Ai}i=0D constitutes a commutative association scheme. In particular, a (strongly) distance-regular digraph Γ is stable, which means that AiT = Ag-i, for each i = 1,…,g — 1, where g denotes the girth of Γ. 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 Γ 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.