Nilpotent groups whose Difference graphs have positive genus

TL;DR

This paper investigates the topological properties of the difference graph derived from power and enhanced power graphs of finite nilpotent groups, specifically focusing on those with genus at most 2.

Contribution

It characterizes all finite nilpotent groups whose difference graphs have genus or cross-cap at most 2, providing new insights into their topological graph properties.

Findings

Identifies all nilpotent groups with difference graphs of genus ≤ 2

Provides a complete classification based on topological genus

Connects group structure with graph embedding properties

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 $2$.