Toughness and A{\alpha}-spectral radius in graphs

Abstract

Let $\alpha\in[0,1)$, and let $G$ be a connected graph of order $n$ with $n\geq f(\alpha)$, where $f(\alpha)=6$ for $\alpha\in[0,\frac{2}{3}]$ and $f(\alpha)=\frac{4}{1-\alpha}$ for $\alpha\in(\frac{2}{3},1)$. A graph $G$ is said to be $t$-tough if $|S|\geq tc(G-S)$ for each subset $S$ of $V(G)$ with $c(G-S)\geq2$, where $c(G-S)$ is the number of connected components in $G-S$. The $A_{\alpha}$-spectral radius of $G$ is denoted by $\rho_{\alpha}(G)$. In this paper, it is verified that $G$ is a 1-tough graph unless $G=K_1\vee(K_{n-2}\cup K_1)$ if $\rho_{\alpha}(G)\geq\rho_{\alpha}(K_1\vee(K_{n-2}\cup K_1))$, where $\rho_{\alpha}(K_1\vee(K_{n-2}\cup K_1))$ equals the largest root of $x^{3}-((\alpha+1)n+\alpha-3)x^{2}+(\alpha n^{2}+(\alpha^{2}-\alpha-1)n-2\alpha+1)x-\alpha^{2}n^{2}+(3\alpha^{2}-\alpha+1)n-4\alpha^{2}+5\alpha-3=0$. Further, we present an $A_{\alpha}$-spectral radius condition for a graph to be a $t$-tough graph.