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Computation of the resistance distance and the Kirchhoff index for the two types of claw-free cubic graphs

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  • Sardar, Muhammad Shoaib
  • Pan, Xiang-Feng
  • Xu, Shou-Jun

Abstract

Let G be a simple connected graph with vertex set V(G) and edge set E(G). The resistance distance r(u,v) between two vertices u,v∈V(G) is the net effective resistance between them in the electric network constructed from G by replacing each edge with a unit resistor. This function is known to be a metric on the vertex-set of any graph. The sum of resistance distances between pairs of vertices in a G is called Kirchhoff index and is denoted by Kf(G). In this study, we will compute the resistance distance between pairs of vertices of string of diamonds and ring of diamonds by using some methods from electrical network theory such as series and parallel principles, the principle of elimination, the star-triangle transformation, and the delta-wye transformation. Then we determine the exact formulas for the Kirchhoff index of the string of diamonds and ring of diamonds.

Suggested Citation

  • 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).
    • Handle: RePEc:eee:apmaco:v:473:y:2024:i:c:s0096300324001425
    DOI: 10.1016/j.amc.2024.128670
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    References listed on IDEAS

    as
    1. 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.
    2. R. B. Bapat & Somit Gupta, 2010. "Resistance distance in wheels and fans," Indian Journal of Pure and Applied Mathematics, Springer, vol. 41(1), pages 1-13, February.
    3. 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).
    4. 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.
    5. 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).
    6. 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).
    7. 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.
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