  The power graph $\mathcal{P}(G)$ of a group $G$ is the simple graph with
vertex set $G$ and two vertices are adjacent whenever one of them is a positive
power of the other. In this paper, for a finite noncyclic nilpotent group $G$,
we study the minimum degree $\delta(\mathcal{P}(G))$ of $\mathcal{P}(G)$. Under
some conditions involving the prime divisors of $|G|$ and the Sylow subgroups
of $G$, we identify certain vertices associated with the generators of maximal
cyclic subgroups of $G$ such that $\delta(\mathcal{P}(G))$ is equal to the
degree of one of these vertices. As an application, we obtain
$\delta(\mathcal{P}(G))$ for some classes of finite noncyclic abelian groups
$G$.
