The scattering number s(G) (resp. \(\tau \) -toughness) of a graph \(G=(V,E)\) is defined as \(s(G)=\max \left\{ c(G-S)-|S|\right\} \) (resp. \(\tau (G)=\min \left\{ \frac{\vert S \vert }{c(G-S)-1}\right\} \) ), in which the maximum (resp. minimum) is taken over all proper sets \(S\subseteq V(G)\) , where \(c(G-S)\) denotes the number of components of \(G-S\) . These two parameters are known as the variants of the toughness of a graph. In this paper, we study the scattering number and the \(\tau \) -toughness by the eigenvalues of matrices associated with a graph. For the scattering number, we establish a sufficient condition involving the spectral radius for a graph G with minimum degree \(\delta \) such that \(s(G)\le 1\) , as well as a condition involving Laplacian eigenvalues for a graph with \(s(G)\le 1\) . Additionally, we present a lower bound for s(G) of G in terms of its Laplacian eigenvalues. Our results extend or improve the recent corresponding results. A non-complete graph G is \(\tau \) -tough if \(\tau (G)\ge \tau \) . For \(\tau \) -tough graphs, we establish a sufficient condition involving the size (or the signless Laplacian spectral radius) for a graph to be \(\tau \) -tough. Furthermore, we present two lower bounds for \(\tau (G)\) of G in terms of its Laplacian eigenvalues. As applications, several new results on factors are derived, which improve the related existing results.
