Weakly distance-regular digraphs whose underlying graphs are distance-regular, I

Abstract

Weakly distance-regular digraphs are a natural directed version of distance-regular graphs. In [8], the third author and Suzuki proposed a question when an orientation of a distance-regular graph defines a weakly distance-regular digraph. In this paper, we initiate this project, and classify all commutative weakly distance-regular digraphs whose underlying graphs are Hamming graphs, folded n-cubes and Doob graphs, respectively.