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 l of an expression x=s1⋯sl with each si∈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∈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.