On the Wiener polarity index of unicyclic graphs

Abstract

The Wiener polarity index W_p(G) of a graph G = (V, E) is the number of unordered pairs of vertices {u, v} of G such that the distance d_G(u, v) = 3. In this paper, we first present some operations minimizing (resp. maximizing) the Wiener polarity index of connected graphs. By using such operations, the minimum (resp. maximum) Wiener polarity index of unicyclic graphs of order n and girth g are obtained, and the corresponding extremal graphs are determined, where 3 ≤ g ≤ n. In addition, the unicyclic graphs minimizing the Wiener polarity index W_p(G) among all unicyclic graphs G with n vertices and k pendants (resp. maximum degree Δ) are characterized, where 0 ≤ k ≤ n - 3 (resp. 2 ≤ Δ ≤ n-1).