Weakly distance-regular digraphs

Abstract

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))= h ̃ , |{z ∈ VΓ | (∂(x,z),∂(z,x))= i ̃ and (∂(z,y),∂(y,z))= j ̃ }| depends only on h ̃ , i ̃ and j ̃ . 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.