  The power graph $\mathcal{P}(G)$ of a finite group $G$ is the simple
undirected graph whose vertex set is $G$, in which two distinct vertices are
adjacent if one of them is an integral power of the other. For an integer
$n\geq 2$, let $C_n$ denote the cyclic group of order $n$ and let $r$ be the
number of distinct prime divisors of $n$. The minimum degree
$\delta(\mathcal{P}(C_n))$ of $\mathcal{P}(C_n)$ is known for $r\in\{1,2\}$,
see [18]. For $r\geq 3$, under certain conditions involving the prime divisors
of $n$, we identify at most $r-1$ vertices such that $\delta(\mathcal{P}(C_n))$
is equal to the degree of at least one of these vertices. If $r=3$ or if $n$ is
a product of distinct primes, we are able to identify two such vertices without
any condition on the prime divisors of $n$.
