Commuting Involution Graphs for Certain Exceptional Groups of Lie Type

Abstract

<jats:p>Suppose that <jats:italic>G</jats:italic> is a finite group and <jats:italic>X</jats:italic> is a <jats:italic>G</jats:italic>-conjugacy classes of involutions. The commuting involution graph <jats:inline-formula><jats:alternatives><jats:tex-math>$${\\mathcal {C}}(G,X)$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>C</mml:mi>\n <mml:mo>(</mml:mo>\n <mml:mi>G</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>X</mml:mi>\n <mml:mo>)</mml:mo>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> is the graph whose vertex set is <jats:italic>X</jats:italic> with <jats:inline-formula><jats:alternatives><jats:tex-math>$$x, y \\in X$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>x</mml:mi>\n <mml:mo>,</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mo>∈</mml:mo>\n <mml:mi>X</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> being joined if <jats:inline-formula><jats:alternatives><jats:tex-math>$$x \\ne y$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>x</mml:mi>\n <mml:mo>≠</mml:mo>\n <mml:mi>y</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$xy = yx$$</jats:tex-math><mml:math xmlns:mml= http://www.w3.org/1998/Math/MathML >\n <mml:mrow>\n <mml:mi>x</mml:mi>\n <mml:mi>y</mml:mi>\n <mml:mo>=</mml:mo>\n <mml:mi>y</mml:mi>\n <mml:mi>x</mml:mi>\n </mml:mrow>\n </mml:math></jats:alternatives></jats:inline-formula>. Here for various exceptional Lie type groups of characteristic two we investigate their commuting involution graphs.</jats:p>