On embeddings of the difference graph of the intersection power graph and the power graph

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 intersection power graph of a finite group $G$ is a simple undirected graph with vertex set $G$ and two vertices $x$, $y$ are adjacent if $⟨x⟩\cap ⟨y⟩\ne \{e\}$. The difference graph $\mathcal{D}(G)$ of a finite group $G$ is the difference of the intersection power graph and power graph with all isolated vertices removed. We characterized all the finite nilpotent groups $G$ except $2$-group such that the difference graph is planar. Further, we determine all the finite nilpotent groups whose difference graph has genus at most $2$. Moreover, we prove that there does not exist any finite group whose difference graph is projective planar.