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The Wiener polarity index of a graph is defined as the number of unorderedpairs of vertices at distance three. In recent years, this topological indexwas extensively studied since it has many known applications in chemistry andalso in network theory. In this paper, we generalize the result of Behmaram,Yousefi-Azari, and Ashrafi proved in 2012 for calculating the Wiener polarityindex of a graph. An important advantage of our generalization is that it canbe used for graphs that contain $4$-cycles and also for graphs whose differentcycles have more than one common edge. In addition, using the main result aclosed formula for the Wiener polarity index is derived for phenylenes andrecalculated for catacondensed benzenoid graphs. The catacondensed benzenoidgraphs and phenylenes attaining the extremal values with respect to the Wienerpolarity index are also characterized.

Formula for calculating the Wiener polarity index with applications to benzenoid graphs and phenylenes