Notes on the equitable graph of type II of a finite group

Given a finite group $G$, the equitable graph of type II defined on $G$ is an undirected graph with vertex set $G$, in which two distinct vertices $a$ and $b$ are adjacent if and only if either $0<|o(a)-o(b)|\le min\{o(a),o(b)\}$ or one of $a$ and $b$ is equal to $e$, where $o(a)$ and $o(b)$ are the orders of $a$ and $b$, respectively. In this paper, we observe that every equitable graph of type II of a finite group is a generalized lexicographic product and discuss finite groups $G$ whose equitable graph of type II is $C_n$-free, where $n\ge 3$. As an application, we show that every equitable graph of type II of a finite group is perfect. Finally, we characterize the metric dimension of the equitable graph of type II of a finite group.