On combinatorial structure and algebraic characterizations of distance-regular digraphs

Let = (A) denote a simple strongly connected digraph with vertex set X, diameter D, and let \A₀, A: =A₁, A₂, , AD\ denote the set of distance-i matrices of. Let \Rᵢ\₈=₀D denote a partition of X X, where Rᵢ=\ (x, y) X X (Aᵢ) ₗₘ=1\ (0 i D). The digraph is distance-regular if and only if (X, \Rᵢ\₈=₀D) is a commutative association scheme. In this paper, we describe the combinatorial structure of in the sense of equitable partition, and from it we derive several new algebraic characterizations of such a graph, including the spectral excess theorem for distance-regular digraph. Along the way, we also rediscover all well-known algebraic characterizations of such graphs.