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

• Main Authors: Li Zhang, Jing Zhao, Jia-Bao Liu, Micheal Arockiaraj
• Format: Article
• Language: English
• Published: MDPI AG 2018-11-01
• Series: Mathematics
• Subjects: graph; Laplacian matrix; resistance distance; group inverse
• Online Access: https://www.mdpi.com/2227-7390/6/12/283

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 <i>H</i>-join of graphs <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>&#8230;</mo> <mo>,</mo> <msub> <mi>G</mi> <mi>k</mi> </msub> </mrow> </semantics> </math> </inline-formula>. Recall that <i>H</i> is an arbitrary graph with <inline-formula> <math display="inline"> <semantics> <mrow> <mi>V</mi> <mo>(</mo> <mi>H</mi> <mo>)</mo> <mo>=</mo> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>&#8230;</mo> <mo>,</mo> <mi>k</mi> <mo>}</mo> </mrow> </semantics> </math> </inline-formula>, and <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>&#8230;</mo> <mo>,</mo> <msub> <mi>G</mi> <mi>k</mi> </msub> </mrow> </semantics> </math> </inline-formula> are disjoint graphs. Then, the <i>H</i>-join of graphs <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>&#8230;</mo> <mo>,</mo> <msub> <mi>G</mi> <mi>k</mi> </msub> </mrow> </semantics> </math> </inline-formula>, denoted by <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>⋁</mo> <mi>H</mi> </msub> <mrow> <mo>{</mo> <msub> <mi>G</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>&#8230;</mo> <mo>,</mo> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>}</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, is a graph formed by taking <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>G</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>&#8230;</mo> <mo>,</mo> <msub> <mi>G</mi> <mi>k</mi> </msub> </mrow> </semantics> </math> </inline-formula> and joining every vertex of <inline-formula> <math display="inline"> <semantics> <msub> <mi>G</mi> <mi>i</mi> </msub> </semantics> </math> </inline-formula> to every vertex of <inline-formula> <math display="inline"> <semantics> <msub> <mi>G</mi> <mi>j</mi> </msub> </semantics> </math> </inline-formula> whenever <i>i</i> is adjacent to <i>j</i> in <i>H</i>. Here, we first give the Laplacian matrix of <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>⋁</mo> <mi>H</mi> </msub> <mrow> <mo>{</mo> <msub> <mi>G</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>&#8230;</mo> <mo>,</mo> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>}</mo> </mrow> </mrow> </semantics> </math> </inline-formula>, and then give a <inline-formula> <math display="inline"> <semantics> <mrow> <mo>{</mo> <mn>1</mn> <mo>}</mo> </mrow> </semantics> </math> </inline-formula>-inverse <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <msup> <mrow> <mo>(</mo> <msub> <mo>⋁</mo> <mi>H</mi> </msub> <mrow> <mo>{</mo> <msub> <mi>G</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>&#8230;</mo> <mo>,</mo> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mrow> <mo>{</mo> <mn>1</mn> <mo>}</mo> </mrow> </msup> </mrow> </semantics> </math> </inline-formula> or group inverse <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <msup> <mrow> <mo>(</mo> <msub> <mo>⋁</mo> <mi>H</mi> </msub> <mrow> <mo>{</mo> <msub> <mi>G</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>&#8230;</mo> <mo>,</mo> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>}</mo> </mrow> <mo>)</mo> </mrow> <mo>#</mo> </msup> </mrow> </semantics> </math> </inline-formula> of <inline-formula> <math display="inline"> <semantics> <mrow> <mi>L</mi> <mo>(</mo> <msub> <mo>⋁</mo> <mi>H</mi> </msub> <mrow> <mo>{</mo> <msub> <mi>G</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>&#8230;</mo> <mo>,</mo> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>}</mo> </mrow> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. It is well know that, there exists a relationship between resistance distance and entries of <inline-formula> <math display="inline"> <semantics> <mrow> <mo>{</mo> <mn>1</mn> <mo>}</mo> </mrow> </semantics> </math> </inline-formula>-inverse or group inverse. Therefore, we can easily obtain resistance distance in <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mo>⋁</mo> <mi>H</mi> </msub> <mrow> <mo>{</mo> <msub> <mi>G</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>G</mi> <mn>2</mn> </msub> <mo>,</mo> <mo>&#8230;</mo> <mo>,</mo> <msub> <mi>G</mi> <mi>k</mi> </msub> <mo>}</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. In addition, some applications are presented in this paper.