Distance-transitive and distance-regular digraphs

We adopt the definition of a distance-transitive digraph given by Lam. If such a digraph has girth g ≥3 and diameter d , we prove the following results: (1) d=g or g −1 and (2) if d=g , then the digraph is obtained by a known construction from a smaller digraph satisfying d=g −1. Finally we mention an alternative definition of distance transitivity, which in our opinion merits study.