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# Resistance Distance in H-Join of Graphs G1,G2,…,Gk

1
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
2
Department of Mathematics, Loyola College, Chennai 600034, India
*
Author to whom correspondence should be addressed.
Mathematics 2018, 6(12), 283; https://doi.org/10.3390/math6120283
Received: 11 October 2018 / Revised: 18 November 2018 / Accepted: 21 November 2018 / Published: 26 November 2018
(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)
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. View Full-Text
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MDPI and ACS Style

Zhang, L.; Zhao, J.; Liu, J.-B.; Arockiaraj, M. Resistance Distance in H-Join of Graphs G1,G2,,Gk. Mathematics 2018, 6, 283. https://doi.org/10.3390/math6120283

AMA Style

Zhang L, Zhao J, Liu J-B, Arockiaraj M. Resistance Distance in H-Join of Graphs G1,G2,,Gk. Mathematics. 2018; 6(12):283. https://doi.org/10.3390/math6120283

Chicago/Turabian Style

Zhang, Li; Zhao, Jing; Liu, Jia-Bao; Arockiaraj, Micheal. 2018. "Resistance Distance in H-Join of Graphs G1,G2,,Gk" Mathematics 6, no. 12: 283. https://doi.org/10.3390/math6120283

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