On the commutation graph of cyclic $TI$-subgroups in linear groups

We study the commutation graph $\Gamma (A)$ of a cyclic $TI$-subgroup $A$ of order 4 in a finite group $G$ with quasisimple generalized Fitting subgroup $F^*(G)$. It is proved that, if $F^*(G)$ is a linear group, then the graph $\Gamma (A)$ is either a coclique or an edge-regular but not coedge-regular graph.