On the combinatorial structure and algebraic characterizations of distance-regular digraphs

Let $\Gamma =\Gamma \left(A\right)$ denote a simple strongly connected digraph with vertex set X, diameter D, and let $\{A_0,A:=A_1,A_2,\dots ,A_D\}$ denote the set of distance-i matrices of Γ. Let ${\left\{R_i\right\}}_{i=0}^D$ denote a partition of $X\times X$, where $R_i=\left\{\right(x,y)\in X\times X|{\left(A_i\right)}_{xy}=1\}$ $(0\le i\le D)$. In the literature, such a digraph Γ is said to be distance-regular if $(X,{\left\{R_i\right\}}_{i=0}^D)$ is a commutative association scheme. In this paper, we provide a combinatorial definition of a distance-regular digraph in terms of equitable partitions. From this definition, we rediscover all well-known algebraic characterizations of such digraphs, including the above one. We also give several new characterizations, and one of them is the spectral excess theorem for distance-regular digraphs.