On the Wiener index of orientations of graphs

The Wiener index of a strong digraph D is defined as the sum shortest path distances between all ordered pairs vertices. This definition has been extended to digraphs that are not necessarily by defining distance from vertex b 0 if there no in D. Knor et al. (2016) [9] considered (not strong) orientations graphs with maximum index. authors conjectured for given tree T, an orientation T always contains v such every u, either (u,v)-path or (v,u)-path In this paper we disprove conjecture. We also show problem finding graph NP-complete, thus answering question [8]. briefly discuss corresponding minimum graph, and special case deciding on m edges can be solved time quadratic order graph.