Generalized toughness and Q-index in a graph

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 $t(G)=\min \{\frac{\mathrm{\mid }S\mathrm{\mid }}{c(G-S)}:S\subseteq V(G),c(G-S)\ge 2\}$ for $G\mathrm{\ne }{K}_n$ and $t(G)=+\mathrm{\infty }$ 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 $t_l(G)=\min \{\frac{\mathrm{\mid }S\mathrm{\mid }}{c(G-S)}:S\subset V(G),c(G-S)\ge l\}$ if $2\le l\le \alpha (G)$, and $t_l(G)=+\mathrm{\infty }$ if $l>\alpha (G)$. If $t_l(G)\ge t$, 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.