Nilpotent groups whose Difference graphs have positive genus

Abstract

The power graph of a finite group $G$ is a simple undirected graph with vertex set $G$ and two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group $G$ is a simple undirected graph whose vertex set is the group $G$ and two vertices $a$ and $b$ are adjacent if there exists $c \in G$ such that both $a$ and $b$ are powers of $c$. In this paper, we study the difference graph $\mathcal{D}(G)$ of a finite group $G$ which is the difference of the enhanced power graph and the power graph of $G$ with all isolated vertices removed. We characterize all the finite nilpotent groups $G$ such that the genus (or cross-cap) of the difference graph $\mathcal{D}(G)$ is at most $2$.