Toughness and Aα-spectral radius in graphs

Let $\alpha \in [0,1)$, and let $G$ be a connected graph of order $n$ with $n\ge f(\alpha )$, 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 $\mathrm{\mid }S\mathrm{\mid }\ge tc(G-S)$ for each subset $S$ of $V(G)$ with $c(G-S)\ge 2$, 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 ${\rho }_{\alpha }(G)\ge {\rho }_{\alpha }(K_1\vee (K_{n-2}\cup K_1))$, 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.