Metric Dimension of Difference Graph of Finite Groups

TL;DR

This paper investigates the metric dimension of the difference graph derived from various finite groups, providing explicit characterizations and calculations for nilpotent and specific non-nilpotent groups.

Contribution

It characterizes the vertex set of the difference graph for finite nilpotent groups and computes its metric dimension, extending results to certain non-nilpotent groups.

Findings

Metric dimension of difference graphs for finite nilpotent groups is determined.

Explicit formulas for the metric dimension of dihedral, quaternion, and semi-dihedral groups.

Characterization of the vertex set of the difference graph for finite nilpotent groups.

Abstract

The Difference graph $\mathcal{D}(G)$ of a finite group $G$ is the difference of the enhanced power graph $\mathcal{P}_{E}(G)$ and the power graph $\mathcal{P}(G)$ with all the isolated vertices removed. In this paper, we characterize the vertex set of the difference graph of finite nilpotent groups and obtain its cardinality. Consequently, we obtain the metric dimension of the difference graph of finite nilpotent groups. Moreover, this paper determines the metric dimension of the difference graphs of certain non-nilpotent groups, namely: dihedral groups, the generalized quaternion groups, and the semi-dihedral groups.