Journal of Mathematical Chemistry

, Volume 57, Issue 1, pp 370–383 | Cite as

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

  • Niko TratnikEmail author
Original Paper


The Wiener polarity index of a graph is defined as the number of unordered pairs of vertices at distance three. In recent years, this topological index was extensively studied since it has many known applications in chemistry and also in network theory. In this paper, we generalize the result of Behmaram et al. (Appl. Math. Lett. 25:1510–1513, 2012) for calculating the Wiener polarity index of a graph. An important advantage of our generalization is that it can be used for graphs that contain 4-cycles and also for graphs whose different cycles have more than one common edge. In addition, using the main result a closed formula for the Wiener polarity index is derived for phenylenes and recalculated for catacondensed benzenoid graphs. The catacondensed benzenoid graphs and phenylenes attaining the extremal values with respect to the Wiener polarity index are also characterized.


Wiener polarity index Benzenoid graph Phenylene First Zagreb index Second Zagreb index 

Mathematics Subject Classification

92E10 05C12 05C90 



The author was financially supported by the Slovenian Research Agency (research core funding Nos. P1-0297 and J1-9109).


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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Faculty of Natural Sciences and MathematicsUniversity of MariborMariborSlovenia
  2. 2.Institute of Mathematics, Physics and MechanicsLjubljanaSlovenia

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