On the power graph of a finite group

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.