On the Extremal Wiener Polarity Index of Hückel Graphs

Graphs are used to model chemical compounds and drugs. In the graphs, each vertex represents an atom of molecule and edges between the corresponding vertices are used to represent covalent bounds between atoms. The Wiener polarity index 𝑊𝑝(𝐺) of a graph 𝐺 is the number of unordered pairs of vertices 𝑢, V of 𝐺 such that the distance between 𝑢 and V is equal to 3. The trees and unicyclic graphs with perfect matching, of which all vertices have degrees not greater than three, are referred to as the Huckel trees ¨ and unicyclic Huckel graphs ¨ , respectively. In this paper, we first consider the smallest and the largest Wiener polarity index among all Huckel trees on ¨ 2𝑛 vertices and characterize the corresponding extremal graphs. Then we obtain an upper and lower bound for the Wiener polarity index of unicyclic Huckel graphs on ¨ 2𝑛 vertices.