Toughness and Aα-spectral radius in graphs

Abstract

Let $\hat{I}\pm \in [0,1)$, and let $G$ be a connected graph of order $n$ with $n\ge f(\hat{I}\pm )$, where $f(\hat{I}\pm )=6$ for $\hat{I}\pm \in [0,\frac{2}{3}]$ and $f(\hat{I}\pm )=\frac{4}{1-\hat{I}\pm }$ for $\hat{I}\pm \in (\frac{2}{3},1)$. A graph $G$ is said to be $t$-tough if $|S|\ge tc(G-S)$ 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 $A_{\hat{I}}\pm$-spectral radius of $G$ is denoted by $\ddot{I}{}_{\hat{I}}\pm (G)$. 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 $\ddot{I}{}_{\hat{I}}\pm (G)\ge \ddot{I}{}_{\hat{I}}\pm (K_1\vee (K_{n-2}\cup K_1))$, where $\ddot{I}{}_{\hat{I}}\pm (K_1\vee (K_{n-2}\cup K_1))$ equals the largest root of $x^3-((\hat{I}\pm +1)n+\hat{I}\pm -3)x^2+(\hat{I}\pm n^2+(\hat{I}{\pm }^2-\hat{I}\pm -1)n-2\hat{I}\pm +1)x-\hat{I}{\pm }^2{n}^2+(3\hat{I}{\pm }^2-\hat{I}\pm +1)n-4\hat{I}{\pm }^2+5\hat{I}\pm -3=0$. Further, we present an $A_{\hat{I}}\pm$-spectral radius condition for a graph to be a $t$-tough graph.

12 pages