Thin weakly distance-regular digraphs

Weakly distance-regular digraphs are digraphs with structures of association schemes. in this paper, we define a class of thin weakly distance-regular digraphs, and give its classification. For a finite group G and a generating subset S, let \x\(s) denote the minimum length I of an expression x = s(1) ... s(l) with each s(i) is an element of S. The classification of thin weakly distance-regular digraphs amounts to the determination of all finite groups G possessing a distinguished generating subset S that satisfies the following property: For x, y is an element of G, (\x\(s), \x(-1)\(s)) = (\y\(s), \y(-1)\(s)) implies x = y. As an application we also classify weakly distance-regular digraphs of out-degree two. (C) 2004 Elsevier Inc. All rights reserved.