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

Abstract

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 $\mbox{bind}(G)$ of $G$ is defined by $\mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}$. If $\mbox{bind}(G)\geq1$, 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 $o(G-S)\leq|S|$ 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 $n\geq n(\alpha)$ has a perfect matching unless $G=K_1\vee(K_{n-5}\cup K_3\cup K_1)$ if $\rho_{\alpha}(G)\geq\rho_{\alpha}(K_1\vee(K_{n-5}\cup K_3\cup K_1))$, 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 $\alpha=\frac{1}{2}$.