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Resistance Distance in H -Join of Graphs G 1 , G 2 , … , G k

Author

Listed:
  • Li Zhang

    (School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China)

  • Jing Zhao

    (School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China)

  • Jia-Bao Liu

    (School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China)

  • Micheal Arockiaraj

    (Department of Mathematics, Loyola College, Chennai 600034, India)

Abstract

In view of the wide application of resistance distance, the computation of resistance distance in various graphs becomes one of the main topics. In this paper, we aim to compute resistance distance in H -join of graphs G 1 , G 2 , … , G k . Recall that H is an arbitrary graph with V ( H ) = { 1 , 2 , … , k } , and G 1 , G 2 , … , G k are disjoint graphs. Then, the H -join of graphs G 1 , G 2 , … , G k , denoted by ⋁ H { G 1 , G 2 , … , G k } , is a graph formed by taking G 1 , G 2 , … , G k and joining every vertex of G i to every vertex of G j whenever i is adjacent to j in H . Here, we first give the Laplacian matrix of ⋁ H { G 1 , G 2 , … , G k } , and then give a { 1 } -inverse L ( ⋁ H { G 1 , G 2 , … , G k } ) { 1 } or group inverse L ( ⋁ H { G 1 , G 2 , … , G k } ) # of L ( ⋁ H { G 1 , G 2 , … , G k } ) . It is well know that, there exists a relationship between resistance distance and entries of { 1 } -inverse or group inverse. Therefore, we can easily obtain resistance distance in ⋁ H { G 1 , G 2 , … , G k } . In addition, some applications are presented in this paper.

Suggested Citation

  • Li Zhang & Jing Zhao & Jia-Bao Liu & Micheal Arockiaraj, 2018. "Resistance Distance in H -Join of Graphs G 1 , G 2 , … , G k," Mathematics, MDPI, vol. 6(12), pages 1-10, November.
    • Handle: RePEc:gam:jmathe:v:6:y:2018:i:12:p:283-:d:185609
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Frank Werner, 2019. "Discrete Optimization: Theory, Algorithms, and Applications," Mathematics, MDPI, vol. 7(5), pages 1-4, May.
    2. Li Zhang & Jing Zhao & Jia-Bao Liu & Salama Nagy Daoud, 2019. "Resistance Distance in the Double Corona Based on R -Graph," Mathematics, MDPI, vol. 7(1), pages 1-13, January.

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