Two-point resistances in the generalized phenylenes
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Two-point resistances in the generalized phenylenes Qishun Li1 · Shuchao Li2
· Leilei Zhang2
Received: 13 July 2019 / Accepted: 12 June 2020 © Springer Nature Switzerland AG 2020
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 − 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 is a nontrivial family with diameter going to ∞ 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. Keywords Resistance distance · Generalized phenylenes · Asymptotic property
1 Introduction We start with introducing some background information that will lead to our main results. Some important previously established facts will also be given. Throughout this paper we consider finite, undirected and simple graphs. Let G = (VG , E G ) be a graph with vertex set VG and edge set E G . For u ∈ VG , denote by dG (u) the degree of the vertex u in G.
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Shuchao Li [email protected] Leilei Zhang [email protected]
1
School of Electrical Engineering, Beijing Jiaotong University, Beijing 100044, People’s Republic of China
2
School of Mathematics and Statistics, Huazhong Normal University, Wuhan 430079, People’s Republic of China
123
Journal of Mathematical Chemistry
1.1 Background The resistance distance r G (u, v) between vertices u and v in a graph G is defined as the effective electrical resistance between u and v when unit resistor is placed on every edge of G. It is one important measure of quantifying structural properties for the given graph and it has been extensively studied in mathematical, physical and chemical areas recently [28,37]. It is well-known [5,9,38] that the escape probability, the first passage time, the cover cost and the commute time of random walks have closely relation with the resistance. For more advances one may be referred to [10,11,13,14] and the references cited in. There are general mathematical resistance-distance-based fundamental quantities (or invariants), e.g., centrality [21] or cyclicity [22]. Although to obtain analytical formulae for resistance distance is difficult as usual, a number of well-known techniques are up to now developed, such as series and parallel rules, sum rules [7,20], general − Y transformations [34], where the series and parallel
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