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 $o(G-S)\le |S|$ 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 $n\ge n(\alpha )$ 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 $\alpha =\frac{1}{2}$.

11 pages