Resistance Distance in <i>H</i>-Join of Graphs <i>G</i>₁,<i>G</i>₂,<i>…</i>,<i>G</i>_k

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.

Keywords

• graph
• Laplacian matrix
• resistance distance
• group inverse