Digraphs with large maximum Wiener index

Recently the concept of Wiener index was extended to digraphs which are not-necessarily strongly connected, and it was shown that some fundamental results extend naturally within this concept. This extension could be applicable in the topics of directed large networks, particularly because with this measure, one assigns finite values to the average distance and betweenness centrality of the nodes in a directed network. It is not hard to show that among digraphs on n   vertices, the directed cycle C→n achieves the maximum Wiener index. Next, we investigate digraphs with the second maximum Wiener index. One can consider this problem in the realm of all digraphs or restricted to those obtained by directing undirected graphs, so called antisymmetric digraphs. In both situations, we obtain that such digraphs are constructed from C→n (n ≥ 6) by adding a single arc. We conclude the paper with consideration for possible further works.