On the power graph of a finite group

TL;DR

This paper investigates the structural properties of the power graph of finite groups, proving it is a perfect graph with a complete core, characterizing its structure via cyclic subgroups, and deriving a formula for its metric dimension.

Contribution

It establishes the perfectness of the power graph, characterizes its structure using cyclic subgroups, and provides a closed formula for its metric dimension, including specific computations for cyclic groups.

Findings

Power graph has a transitive orientation and is a perfect graph.

The core of the power graph is a complete graph.

A closed formula for the metric dimension of the power graph is derived.

Abstract

The power graph $\mathcal P_G$ of a finite group $G$ is the graph with the vertex set $G$, where two elements are adjacent if one is a power of the other. We first show that $\mathcal P_G$ has an transitive orientation, so it is a perfect graph and its core is a complete graph. Then we use the poset on all cyclic subgroups (under usual inclusion) to characterise the structure of $\mathcal P_G$. Finally, the closed formula for the metric dimension of $\mathcal P_G$ is established. As an application, we compute the metric dimension of the power graph of a cyclic group.