  The enhanced power graph of a finite group $G$, denoted by
$\mathcal{P}_E(G)$, is the simple undirected graph whose vertex set is $G$ and
two distinct vertices $x, y$ are adjacent if $x, y \in \langle z \rangle$ for
some $z \in G$. In this article, we determine all finite groups such that the
minimum degree and the vertex connectivity of $\mathcal{P}_E(G)$ are equal.
Also, we classify all groups whose (proper) enhanced power graphs are strongly
regular. Further, the vertex connectivity of the enhanced power graphs
associated to some nilpotent groups is obtained. Finally, we obtain a lower
bound and an upper bound for the Wiener index of $\mathcal{P}_E(G)$, where $G$
is a nilpotent group. The finite nilpotent groups attaining these bounds are
also characterized.
