Weakly distance-regular digraphs

We consider the following generalization of distance-regular digraphs. A connected digraph F is said to be weakly distance-regular if, for all vertices x and y, with (partial derivative(x, y), partial derivative(y, x)) = (h) over tilde, \{z is an element of VGamma \ (partial derivative)(x,z), partial derivative(z,x)) = (i) over tilde and (partial derivative(z,y),partial derivative(y,z)) =(j) over tilde}\ depends only on (h) over tilde, (i) over tilde and (j) over tilde. We give some constructions of weakly distance-regular digraphs and discuss the connections to association schemes. Finally, we determine all commutative weakly distance-regular digraphs of valency 2. (C) 2002 Elsevier Science B.V. All rights reserved.