  In 1995, Brouwer proved that the toughness of a connected $k$-regular graph
$G$ is at least $k/\lambda-2$, where $\lambda$ is the maximum absolute value of
the non-trivial eigenvalues of $G$. Brouwer conjectured that one can improve
this lower bound to $k/\lambda-1$ and that many graphs (especially graphs
attaining equality in the Hoffman ratio bound for the independence number) have
toughness equal to $k/\lambda$. In this paper, we improve Brouwer's spectral
bound when the toughness is small and we determine the exact value of the
toughness for many strongly regular graphs attaining equality in the Hoffman
ratio bound such as Lattice graphs, Triangular graphs, complements of
Triangular graphs and complements of point-graphs of generalized quadrangles.
For all these graphs with the exception of the Petersen graph, we confirm
Brouwer's intuition by showing that the toughness equals $k/(-\lambda_{min})$,
where $\lambda_{min}$ is the smallest eigenvalue of the adjacency matrix of the
graph.
