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Computation of resistance distance and Kirchhoff index of the two classes of silicate networks

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  • Sardar, Muhammad Shoaib
  • Pan, Xiang-Feng
  • Xu, Si-Ao

Abstract

The resistance distance between two vertices of a simple connected graph G is equal to the resistance between two equivalent points on an electrical network, constructed so as to correspond to G, with each edge being replaced by a unit resistor. The Kirchhoff index of G is the sum of resistance distances between all pairs of vertices in G. In this paper, the resistance distance between any two arbitrary vertices of a chain silicate network and a cyclic silicate network was procured by utilizing techniques from the theory of electrical networks, i.e., the series and parallel principles, the principle of elimination, the star-triangle transformation and the delta-wye transformation. Two closed formulae for the Kirchhoff index of the chain silicate network and the cyclic silicate network were obtained respectively.

Suggested Citation

  • Sardar, Muhammad Shoaib & Pan, Xiang-Feng & Xu, Si-Ao, 2020. "Computation of resistance distance and Kirchhoff index of the two classes of silicate networks," Applied Mathematics and Computation, Elsevier, vol. 381(C).
    • Handle: RePEc:eee:apmaco:v:381:y:2020:i:c:s0096300320302526
    DOI: 10.1016/j.amc.2020.125283
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    References listed on IDEAS

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    1. Sardar, Muhammad Shoaib & Hua, Hongbo & Pan, Xiang-Feng & Raza, Hassan, 2020. "On the resistance diameter of hypercubes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    2. Fan, Jiaqi & Zhu, Jiali & Tian, Li & Wang, Qin, 2020. "Resistance Distance in Potting Networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 540(C).
    3. Jiang, Zhuozhuo & Yan, Weigen, 2017. "Resistance between two nodes of a ring network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 484(C), pages 21-26.
    4. Zhang, Teng & Bu, Changjiang, 2019. "Detecting community structure in complex networks via resistance distance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 526(C).
    5. Liu, Jia-Bao & Pan, Xiang-Feng, 2016. "Minimizing Kirchhoff index among graphs with a given vertex bipartiteness," Applied Mathematics and Computation, Elsevier, vol. 291(C), pages 84-88.
    6. Huang, Guixian & He, Weihua & Tan, Yuanyao, 2019. "Theoretical and computational methods to minimize Kirchhoff index of graphs with a given edge k-partiteness," Applied Mathematics and Computation, Elsevier, vol. 341(C), pages 348-357.
    7. Fei, Junqi & Tu, Jianhua, 2018. "Complete characterization of bicyclic graphs with the maximum and second-maximum degree Kirchhoff index," Applied Mathematics and Computation, Elsevier, vol. 330(C), pages 118-124.
    8. Tu, Jianhua & Du, Junfeng & Su, Guifu, 2015. "The unicyclic graphs with maximum degree resistance distance," Applied Mathematics and Computation, Elsevier, vol. 268(C), pages 859-864.
    9. Yang, Yujun & Cao, Yuliang & Yao, Haiyuan & Li, Jing, 2018. "Solution to a conjecture on a Nordhaus–Gaddum type result for the Kirchhoff index," Applied Mathematics and Computation, Elsevier, vol. 332(C), pages 241-249.
    10. Ma, Xiaoling & Bian, Hong, 2019. "The normalized Laplacians, degree-Kirchhoff index and the spanning trees of hexagonal Möbius graphs," Applied Mathematics and Computation, Elsevier, vol. 355(C), pages 33-46.
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    Cited by:

    1. Sajjad, Wasim & Sardar, Muhammad Shoaib & Pan, Xiang-Feng, 2024. "Computation of resistance distance and Kirchhoff index of chain of triangular bipyramid hexahedron," Applied Mathematics and Computation, Elsevier, vol. 461(C).
    2. Sun, Wensheng & Yang, Yujun, 2023. "Extremal pentagonal chains with respect to the Kirchhoff index," Applied Mathematics and Computation, Elsevier, vol. 437(C).
    3. Sardar, Muhammad Shoaib & Pan, Xiang-Feng & Xu, Shou-Jun, 2024. "Computation of the resistance distance and the Kirchhoff index for the two types of claw-free cubic graphs," Applied Mathematics and Computation, Elsevier, vol. 473(C).
    4. Lin, Wei & Li, Min & Zhou, Shuming & Liu, Jiafei & Chen, Gaolin & Zhou, Qianru, 2021. "Phase transition in spectral clustering based on resistance matrix," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 566(C).

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