| Summary: | 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>∈</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>.
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