Extremal Estimates of the Wiener Index for Weakly Connected Directed Graphs

This article considers the Wiener index for weakly connected directed graphs. For such graphs, the distance $$d(u,v)$$ between vertices $$u$$ and $$v$$ is not always defined, which requires a correction for the Wiener index to be meaningful. The convention where it is assumed that $$d(u,v) = 0$$ in the absence of a path between vertices is well-studied. We consider the convention where $$d(u,v)$$ is equal to the number of vertices in the graph when there is no path between vertices $$u$$ and $$v$$ . This article presents graphs with $$n$$ vertices for which the Wiener index with this convention reaches the minimal and maximal values. We also present experimental results showing how the Wiener index (considering both conventions of distance) changes when arcs are added to a weakly connected directed graph with fixed and random structures.