On combinatorial structure and algebraic characterizations of distance-regular digraphs

Abstract: 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.