A note on commuting graphs for general linear groups

Abstract

Let $G$ be a group and $X$ a subset of $G$. Then $\mathcal{C}(G, X)$ is a graph with vertex set $X$ in which two distinct elements $x$, $y\in X$ are joined by an edge if $xy=yx$. In this paper, we study the clique number, the domination number, the diameter, the planarity, the perfection and regularity of $\mathcal{C}(G, X)$ where $G=GL(n,q)$ and $X$ is the set of transvections.