Weakly distance-regular digraphs Original Research Article

We consider the following generalization of distance-regular digraphs. A connected digraph Γ is said to be weakly distance-regular if, for all vertices x and y with (∂(x,y),∂(y,x))=h̃, |{z ∈ VΓ | (∂(x,z),∂(z,x))=ĩ and (∂(z,y),∂(y,z))=j̃}| depends only on h̃, ĩ and j̃. 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.