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Two-point resistances in the generalized phenylenes

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Abstract

The resistance between two nodes in some electronic networks has been studied extensively. Let \(G_n\) be a generalized phenylene with n 6-cycles and n 4-cycles. Using series and parallel rules and the \(\Delta - Y\) transformations we obtain explicit formulae for the resistance distance between any two points of \(G_n\). To the best of our knowledge \(\{G_n\}_{n=1}^{\infty }\) is a nontrivial family with diameter going to \(\infty \) for which all resistance distances have been explicitly calculated. We also determine the maximal resistance distance and the minimal resistance distance in \(G_n\). The monotonicity and some asymptotic properties of resistance distances in \(G_n\) are given. At last some numerical results are discussed, in which we calculate the Kirchhoff indices of a set of benzenoid hydrocarbons; We compare their Kirchhoff indices with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the Kirchhoff index is better than or as good as the models corresponding to the other distance-based indices.

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Acknowledgements

Our thanks are due to the referee and Professor D. J. Klein for careful reading of this manuscript of this paper, and for suggesting may improvements.

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  1. School of Electrical Engineering, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Qishun Li

  2. School of Mathematics and Statistics, Huazhong Normal University, Wuhan, 430079, People’s Republic of China

    Shuchao Li & Leilei Zhang

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Li, Q., Li, S. & Zhang, L. Two-point resistances in the generalized phenylenes. J Math Chem 58, 1846–1873 (2020). https://doi.org/10.1007/s10910-020-01152-z

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