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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 W    p  ( 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  Hückel trees  and  unicyclic Hückel graphs , respectively. In this paper, we first consider the smallest and the largest Wiener polarity index among all Hückel trees on 2 n  vertices and characterize the corresponding extremal graphs. Then we obtain an upper and lower bound for the Wiener polarity index of unicyclic Hückel graphs on 2 n  vertices.

On the Extremal Wiener Polarity Index of Hückel Graphs