  Let $\Gamma=\Gamma(A)$ denote a simple strongly connected digraph with vertex
set $X$, diameter $D$, and let $\{A_0,A:=A_1,A_2,\ldots,A_D\}$ denote the set
of distance-$i$ matrices of $\Gamma$. 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 $\Gamma$ 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 $\Gamma$ 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.
