Some remarks on Wiener index of oriented graphs

In this paper, we study the Wiener index (i.e., the total distance or the transmission number) of not necessarily strongly connected digraphs. In order to do so, if there is no directed path from u to v  , we follow the convention d(u,v)=0,d(u,v)=0, which was independently introduced in several studies of directed networks. Under this assumption we naturally generalize the Wiener theorem, as well as a relation between the Wiener index and betweenness centrality to directed graphs. We formulate and study conjectures about orientations of undirected graphs which achieve the extremal values of Wiener index.