Let \G=\G(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 \G. Let \{R_i\}_{i=0}^D denotes 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 \G 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 \G in the sense of equitable partition and from it we derive several algebraic characterizations of such a graph.

Among else, we prove that the following (i)–(iii) are equivalent.
(i) \G is a distance-regular digraph.
(ii) \G is regular, has spectrally maximum diameter (D = d) and both matrices A^\top and the distance-D matrix A_D are polynomial in A.
(iii) (X,\{R_{\ii}\}_{\ii\in\Delta}) is a commutative D-class association scheme, where \Delta=\{(\partial(x,y),\partial(y,x))\mid x,y\in X\}, and for any \ii\in\Delta, R_{\ii} denote the set of ordered pairs (x,y)\in X\times X such that (\partial(x,y),\partial(y,x))=\ii (here \partial(x,y) denote the distance from vertex x to vertex y).

This is work in progress (it is joint work with Giusy Monzillo).

We look forward to seeing you there!