On combinatorial structure and algebraic characterizations of distance-regular digraphs

Let $\mathrm{\Gamma }=\mathrm{\Gamma }(A)$ 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 $\mathrm{\Gamma }$. Let $\{R_i{\}}_{i=0}^D$ denote a partition of $X\times X$, where $R_i=\{(x,y)\in X\times X\mid (A_i{)}_{xy}=1\}$ $(0\le i\le D)$. The digraph $\mathrm{\Gamma }$ is distance-regular if and only if $(X,\{R_i{\}}_{i=0}^D)$ is a commutative association scheme. In this paper, we describe the combinatorial structure of $\mathrm{\Gamma }$ 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.