On the maximal $\alpha$-spectral radius of graphs with given matching number

Let $\mathscr{G}_{n,\beta}$ be the set of graphs of order $n$ with given matching number $\beta$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue of the nonnegative matrix $A_{\alpha}(G)=\alpha D(G)+A(G)$ is called the $\alpha$-spectral radius of $G$. The graphs with maximal $\alpha$-spectral radius in $\mathscr{G}_{n,\beta}$ are completely characterized in this paper. In this way we provide a general framework to attack the problem of extremal spectral radius in $\mathscr{G}_{n,\beta}$. More precisely, we generalize the known results on the maximal adjacency spectral radius in $\mathscr{G}_{n,\beta}$ and the signless Laplacian spectral radius.