On some properties of enhanced power graph

Given a group $G$, the enhanced power graph of $G$ denoted by $\mathcal{G}_e(G)$, is the graph with vertex set $G$ and two distinct vertices $x, y$ are edge connected in $\mathcal{G}_e(G)$ if there exists $z\in G$ such that $x=z^m$ and $y=z^n$, for some $m, n\in \mathbb{N}$. In this article, we characterize the enhanced power graph $\mathcal{G}_e(G)$ of $G$. The graph $\mathcal{G}_e(G)$ is complete if and only if $G$ is cyclic, and $\mathcal{G}_e(G)$ is Eulerian if and only if $|G|$ is odd. We classify all abelian groups and also all non-abelian $p-$groups $G$ for which $\mathcal{G}_e(G)$ satisfies the cone property.