  In 1977, Hoffman gave a characterization of graphs with smallest eigenvalue
at least $-2$. In this paper we generalize this result to graphs with smaller
smallest eigenvalue. For the proof, we use a combinatorial object named Hoffman
graph, introduced by Woo and Neumaier in 1995. Our result says that for every
$\lambda \leq -2$, if a graph with smallest eigenvalue at least $\lambda$
satisfies some local conditions, then it is highly structured. We apply our
result to graphs which are cospectral with the Hamming graph $H(3,q)$, the
Johnson graph $J(v, 3)$ and the $2$-clique extension of grids, respectively.
