Commutative weakly distance-regular digraphs of girth 2

We introduce some constructions of weakly distance-regular digraphs of girth 2, and prove that a certain quotient digraph of a commutative weakly distance-transitive digraph of girth 2 is a distance-transitive graph. As an application of the result, we obtain some examples of weakly distance-regular digraphs which are not weakly distance-transitive. Moreover, a class of commutative weakly distance-regular (respectively weakly distance-transitive) digraphs of girth 2 is characterized. Finally, all commutative weakly distance-regular digraphs of valency 3 and girth 2 are classified.