Generalized toughness and Q-index in a graph

Abstract

Let $G$ be a graph. We denote by $c(G)$, $\alpha(G)$ and $q(G)$ the number of components, the independence number and the signless Laplacian spectral radius ($Q$-index for short) of $G$, respectively. The toughness of $G$ is defined by $t(G)=\min\left\{\frac{|S|}{c(G-S)}:S\subseteq V(G), c(G-S)\geq2\right\}$ for $G\neq K_n$ and $t(G)=+\infty$ for $G=K_n$. Chen, Gu and Lin [Generalized toughness and spectral radius of graphs, Discrete Math. 349 (2026) 114776] generalized this notion and defined the $l$-toughness $t_l(G)$ of a graph $G$ as $t_l(G)=\min\left\{\frac{|S|}{c(G-S)}:S\subset V(G), c(G-S)\geq l\right\}$ if $2\leq l\leq\alpha(G)$, and $t_l(G)=+\infty$ if $l>\alpha(G)$. If $t_l(G)\geq t$, then $G$ is said to be $(t,l)$-tough. In this paper, we put forward $Q$-index conditions for a graph to be $(b,l)$-tough and $(\frac{1}{b},l)$-tough, respectively.