Finite groups whose commuting graphs are line graphs

Abstract

The commuting graph $\hat{I}“(G)$ of a group $G$ is the simple undirected graph with group elements as a vertex set and two elements $x$ and $y$ are adjacent if and only if $xy=yx$ in $G$. By eliminating the identity element of $G$ and all the dominant vertices of $\hat{I}“(G)$, the resulting subgraphs of $\hat{I}“(G)$ are $\hat{I}{“}^{\ast }(G)$ and $\hat{I}{“}^{\ast \ast }(G)$, respectively. In this paper, we classify all the finite groups $G$ such that the graph $\hat{I}”(G)\in \{\hat{I}“(G),\hat{I}{“}^{\ast }(G),\hat{I}{“}^{\ast \ast }(G)\}$ is the line graph of some graph. We also classify all the finite groups $G$ whose graph $\hat{I}”(G)\in \{\hat{I}“(G),\hat{I}{“}^{\ast }(G),\hat{I}{“}^{\ast \ast }(G)\}$ is the complement of line graph.