Commuting Graphs, <i>C</i>(<i>G</i>, <i>X</i>) in Symmetric Groups Sym(<i>n</i>) and Its Connectivity

• Main Authors: Athirah Nawawi, Sharifah Kartini Said Husain, Muhammad Rezal Kamel Ariffin
• Format: Article
• Language: English
• Published: MDPI AG 2019-09-01
• Series: Symmetry
• Subjects: commuting graph; symmetric group; order 3 elements
• Online Access: https://www.mdpi.com/2073-8994/11/9/1178

A commuting graph is a graph denoted by <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">C</mi> <mo>(</mo> <mi>G</mi> <mo>,</mo> <mi>X</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> where <i>G</i> is any group and <i>X</i>, a subset of a group <i>G</i>, is a set of vertices for <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">C</mi> <mo>(</mo> <mi>G</mi> <mo>,</mo> <mi>X</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. Two distinct vertices, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>&#8712;</mo> <mi>X</mi> </mrow> </semantics> </math> </inline-formula>, will be connected by an edge if the commutativity property is satisfied or <inline-formula> <math display="inline"> <semantics> <mrow> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mi>y</mi> <mi>x</mi> </mrow> </semantics> </math> </inline-formula>. This study presents results for the connectivity of <inline-formula> <math display="inline"> <semantics> <mrow> <mi mathvariant="script">C</mi> <mo>(</mo> <mi>G</mi> <mo>,</mo> <mi>X</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> when <i>G</i> is a symmetric group of degree <i>n</i>, Sym<inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, and <i>X</i> is a conjugacy class of elements of order three in <i>G</i>.