  A graph $G$ is said to be $k$-extendable if every matching of size $k$ in $G$
can be extended to a perfect matching of $G$, where $k$ is a positive integer.
We say $G$ is $1$-excludable if for every edge $e$ of $G$, there exists a
perfect matching excluding $e$. In this paper, we first establish a lower bound
on the size (resp. the spectral radius) of $G$ to guarantee that $G$ is
$k$-extendable. Then we determine a lower bound on the size (resp. the spectral
radius) of $G$ to guarantee that $G$ is $1$-excludable. All the corresponding
extremal graphs are characterized.
