On average distance in tournaments and Eulerian digraphs

We present upper bounds on the average distance for two important classes of digraphs, viz., tournaments and Eulerian digraphs. We first show that the average distance of Eulerian digraphs of order n and minimum degree δ is bounded from above by [Formula presented]. The coefficient [Formula presented] is close to being optimal. We also give an improved bound for Eulerian bipartite digraphs. We then give upper bounds on the average distance of tournaments in terms of order and edge-connectivity, and in terms of diameter only. Both bounds are sharp apart from an additive constant.