Perfect k-matching, k-factor-critical and Aα-spectral radius

A $k$-matching of a graph $G$ is a function $f$: $E\left(G\right)\to \{0,1,2,\dots ,k\}$ satisfying $\sum _{e\in \partial \left(v\right)}f\left(e\right)\le k$ for any vertex $v\in V\left(G\right)$. A $k$-matching of a graph $G$ is perfect if $\sum _{e\in \partial \left(v\right)}f\left(e\right)=k$ for every vertex $v\in V\left(G\right)$. A graph $G$ of order $n$ is k-factor-critical if the removal of any set of $k$ vertices of $G$ results in a graph with a perfect matching. Let $A\left(G\right)$ and $D\left(G\right)$ be the adjacency matrix and the degree diagonal matrix of $G$. For $\alpha \in [0,1]$, Nikiforov $\left(2017\right)$ introduced the $A_{\alpha }$-matrix of G as follows: $A_{\alpha }\left(G\right)=\alpha D\left(G\right)+(1-\alpha )A\left(G\right).$ In this paper, according to the $A_{\alpha }$-spectral radius, we provide two sufficient conditions to ensure that a graph is $k$-factor-critical and has a perfect $k$-matching, respectively.