Let $\alpha \in [0,1)$, and let G be a graph of even order n with $n\ge f(\alpha )$, where $f(\alpha )=10$ for $0\le \alpha \le 1/2$, $f(\alpha )=14$ for $1/2<\alpha \le 2/3$ and $f(\alpha )=5/(1-\alpha )$ for $2/3<\alpha <1$. In this paper, it is shown that if the $A_{\alpha }$-spectral radius of G is not less than the largest root of $x^3-((\alpha +1)n+\alpha -4)x^2+(\alpha n^2+({\alpha }^2-2\alpha -1)n-2\alpha +1)x-{\alpha }^2{n}^2+(5{\alpha }^2-3\alpha +2)n-10{\alpha }^2+15\alpha -8=0$ then G has a perfect matching unless $G=K_1\mathrm{\nabla }(K_{n-3}\cup 2K_1)$. This generalizes a result of S. O (2021) [18], which gives a sufficient condition for the existence of a perfect matching in a graph in terms of the adjacency spectral radius.

中文翻译：

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Aα-谱半径和图的完美匹配

让 $\alpha \in [0,1)$，让G是一个偶数阶n的图，其中$n\ge F(\alpha )$， 在哪里 $F(\alpha )=10$ 为了 $0\le \alpha \le 1/2$, $F(\alpha )=14$ 为了 $1/2<\alpha \le 2/3$ 和 $F(\alpha )=5/(1-\alpha )$ 为了 $2/3<\alpha <1$. 在本文中，表明如果${\mathrm{一种}}_{\alpha }$的-spectral半径ģ不大于最大根少$X^3-((\alpha +1)n+\alpha -4)X^2+(\alpha n^2+({\alpha }^2-2\alpha -1)n-2\alpha +1)X-{\alpha }^2{n}^2+(5{\alpha }^2-3\alpha +2)n-10{\alpha }^2+15\alpha -8=0$那么G有一个完美的匹配，除非$G={钾}_1\mathrm{\nabla }({钾}_{n-3}\cup 2{钾}_1)$. 这概括了 S. O (2021) [18] 的结果，该结果为在邻接谱半径方面的图中存在完美匹配提供了充分条件。