Year: 2026
Type: preprint
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 $\langle x\rangle \cap \langle y \rangle \neq \{e\}$. 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.
Source: arXiv (Cornell University)
Authors: Manisha, Ekta, Kumar, Jitender
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7208725378