Perfect matchings and $A_{\alpha }$-spectral radius in 1-binding graphs

Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. For $\alpha \in [0,1)$, we use $A_{\alpha }(G)$ and ${\rho }_{\alpha }(G)$ to denote the $A_{\alpha }$-matrix and the $A_{\alpha }$-spectral radius of $G$, respectively. The binding number $bind(G)$ of $G$ is defined by $bind(G)=\min \{\frac{\mathrm{\mid }{N}_G(X)\mathrm{\mid }}{\mathrm{\mid }X\mathrm{\mid }}:\mathrm{\varnothing }\mathrm{\ne }X\subseteq V(G),N_G(X)\mathrm{\ne }V(G)\}$. If $bind(G)\ge 1$, then $G$ is called 1-binding. A perfect matching in $G$ is a set of nonadjacent edges covering every vertex of $G$. Tutte proved that a graph $G$ of even order has a perfect matching if and only if $o(G-S)\le \mathrm{\mid }S\mathrm{\mid }$ holds for every $S\subseteq V(G)$ [W. Tutte, The factorization of linear graphs, J. Lond. Math. Soc. 22 (1947) 107–111]. In this paper, we use Tutte’s result to prove that a connected 1-binding graph $G$ of even order $n$ with $n\ge n(\alpha )$ has a perfect matching unless $G=K_1\vee (K_{n-5}\cup K_3\cup K_1)$ if ${\rho }_{\alpha }(G)\ge {\rho }_{\alpha }(K_1\vee (K_{n-5}\cup K_3\cup K_1))$, where $n(\alpha )$ is defined as follows: $n(\alpha )=\max \{18,\frac{2+8\alpha }{1-2\alpha }\}$ if $\alpha \in [0,\frac{1}{2})$, and $n(\alpha )=18$ if $\alpha =\frac{1}{2}$.