  The toughness $t(G)$ of a graph $G=(V,E)$ is defined as
$t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum is taken over all
$S\subset V$ such that $G-S$ is disconnected, where $c(G-S)$ denotes the number
of components of $G-S$. We present two tight lower bounds for $t(G)$ in terms
of the Laplacian eigenvalues and provide strong support for a conjecture for a
better bound which, if true, implies both bounds, and improves and generalizes
known bounds by Alon, Brouwer, and the first author. As applications, several
new results on perfect matchings, factors and walks from Laplacian eigenvalues
are obtained, which leads to a conjecture about Hamiltonicity and Laplacian
eigenvalues.
