On the enhanced power graph of a group

Abstract

The enhanced power graph $\mathcal{P}_e(G)$ of a group $G$ is a graph with vertex set $G$ and two vertices are adjacent if they belong to the same cyclic subgroup. In this paper, we consider the minimum degree, independence number and matching number of enhanced power graphs of finite groups. We first study these graph invariants for $\mathcal{P}_e(G)$ when $G$ is any finite group, and then determine them when $G$ is a finite abelian $p$-group, $U_{6n} = \langle a, b : a^{2n} = b^3 = e, ba =ab^{-1} \rangle$, the dihedral group $D_{2n}$, or the semidihedral group $SD_{8n}$. If $G$ is any of these groups, we prove that $\mathcal{P}_e(G)$ is perfect and then obtain its strong metric dimension. Additionally, we give an expression for the independence number of $\mathcal{P}_e(G)$ for any finite abelian group $G$. These results along with certain known equalities yield the edge connectivity, vertex covering number and edge covering number of enhanced power graphs of the respective groups as well.