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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 aperfect matching excluding $e$. In this paper, we first establish a lower boundon 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 spectralradius) of $G$ to guarantee that $G$ is $1$-excludable. All the correspondingextremal graphs are characterized.

Matching extension and matching exclusion via the size or the spectral radius of graphs