On the Wiener Polarity Index

The Wiener polarity index W-P(G) of a graph G is the number of unordered pairs of vertices {u, v} of G such that the distance of u and v is equal to 3. In this paper, we obtain the relation between Wiener polarity index and Zegreb indices, and the relation between Wiener polarity index and Wiener index (resp. hyper-Wiener index). Moreover, we determine the second smallest Wiener polarity index together with the corresponding graphs among all trees on n vertices, we also identify the smallest and the second smallest Wiener polarity indices together with the corresponding graphs, respectively, among all unicyclic graphs on n vertices.