The Line Graph of a Commuting Graph on the Dihedral Group $D_{2n}$

Abstract

Let $\Gamma$ be a non-abelian group and $\alpha \subseteq \Gamma.$ Then the Commuting graph $C\left(\Gamma,\alpha\right)$ has $\alpha$ as its vertex set and two distinct vertices in $\alpha$ are adjacent if they commute with each other in $\Gamma.$ Let $G=L\left(C\left(\Gamma,\alpha\right)\right)$ be the Line graph of the Commuting graph. A vertex $v_{i}$ of $G$ is given by $\left\{x,y\right\}$ = $\left\{y,x\right\}$ where $x$ and $y$ are the vertices that are adjacent in $C\left(\Gamma,\alpha\right).$ In this paper, we discuss certain properties of the Line graph of the Commuting graph on the Dihedral group $D_{2n}.$ More specifically, we obtain the chromatic number, clique number and genus of this graph.