Generalized toughness and Q-index in a graph

TL;DR

This work ties generalized toughness $t_l(G)$ to spectral properties via the signless Laplacian $Q$-index $q(G)$. By developing and employing equitable-partition techniques and known lemmas on subgraphs and $Q$-index bounds, the authors derive two spectral sufficient conditions: one guaranteeing $(b,l)$-toughness and another guaranteeing $(\frac{1}{b},l)$-toughness, under explicit order constraints and extremal-join graph forms. The main contributions are Theorems 1.1 and 1.2, which provide sharp-looking $Q$-index thresholds tied to specific join-graph structures, thereby bridging generalized toughness with spectral criteria. This offers practical, verifiable conditions to certify generalized toughness from spectral data, enriching the interplay between graph toughness and signless Laplacian spectra.

Abstract

Let $G$ be a graph. We denote by $c(G)$, $α(G)$ 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 $2\leq l\leqα(G)$, and $t_l(G)=+\infty$ if $l>α(G)$. 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.