  For a group $G,$ the enhanced power graph of $G$ is a graph with vertex set
$G$ in which two distinct elements $x, y$ are adjacent if and only if there
exists an element $w$ in $G$ such that both $x$ and $y$ are powers of $w.$ The
proper enhanced power graph is the induced subgraph of the enhanced power graph
on the set $G \setminus S,$ where $S$ is the set of dominating vertices of the
enhanced power graph. In this paper, we first characterize the dominating
vertices of enhanced power graph of any finite nilpotent group. Thereafter, we
classify all nilpotent groups $G$ such that the proper enhanced power graphs
are connected and find out their diameter. We also explicitly find out the
domination number of proper enhanced power graphs of finite nilpotent groups.
Finally, we determine the multiplicity of the Laplacian spectral radius of the
enhanced power graphs of nilpotent groups.
