The diameter and dominating sets of the difference graph of a nilpotent group

TL;DR

This paper investigates the difference graph of finite groups, especially nilpotent groups, establishing an upper bound on its diameter and classifying groups based on this diameter.

Contribution

It introduces bounds on the diameter of the difference graph for nilpotent groups and classifies groups according to their difference graph's diameter.

Findings

Diameter of the difference graph of a nilpotent group is at most 4.

Classification of nilpotent groups with difference graph diameter k for each k ≤ 4.

Generalization and refinement of previous results by Biswas et al.

Abstract

Given a finite group $G$, the difference graph of $G$, denoted by $\mathcal{D}(G)$, is the difference of the enhanced power graph of $G$ and the power graph of $G$, with all isolated vertices removed. This paper mainly studies the dominating sets of the difference graph of a finite group. In particular, we prove that the diameter of the difference graph of a nilpotent group has an upper bound of $4$. Furthermore, we generalize and refine the result by Biswas et al. by classifying all nilpotent groups whose difference graph has diameter $k$, for each $k\le 4$.