Toughness and signless Laplacian eigenvalues in graphs

Abstract

For a non-complete connected graph $G$, the toughness of $G$ is defined as $t(G)=\min\Big\{\frac{|S|}{c(G-S)}\Big\}$ by Chvátal, in which the minimum is taken over all proper sets $S\subset V(G)$, where $c(G-S)$ denotes the number of components of $G-S$ and $c(G-S)\ge2$. A graph $G$ is called $t$-tough if $t(G)\geq t$. Incorporating the toughness and signless Laplacian eigenvalues of a regular graph, we provide a sufficient spectral radius condition for a connected regular graph to be $\frac{1}{b}$-tough, where $b\geq1$ is an integer. Particularly, we obtain the second largest eigenvalue condition for regular graphs to be $\frac{1}{b}$-tough. Moreover, we give a sufficient condition for a connected graph to be $\frac{1}{b}$-tough with fixed minimum degree $\delta$ in terms of the signless Laplacian spectral radius and spectral radius, which extends a significant result by Fan, Lin and Lu [Eur. J. Comb. 110:Paper No. 103701, 2023].