  The smallest eigenvalue of a graph is the smallest eigenvalue of its
adjacency matrix. We show that the family of graphs with smallest eigenvalue at
least $-\lambda$ can be defined by a finite set of forbidden induced subgraphs
if and only if $\lambda < \lambda^*$, where $\lambda^* = \rho^{1/2} +
\rho^{-1/2} \approx 2.01980$, and $\rho$ is the unique real root of $x^3 = x +
1$. This resolves a question raised by Bussemaker and Neumaier. As a byproduct,
we find all the limit points of smallest eigenvalues of graphs, supplementing
Hoffman's work on those limit points in $[-2, \infty)$.
  We also prove that the same conclusion about forbidden subgraph
characterization holds for signed graphs. Our impetus for the study of signed
graphs is to determine the maximum cardinality of a spherical two-distance set
with two fixed angles (one acute and one obtuse) in high dimensions. Denote by
$N_{\alpha, \beta}(n)$ the maximum number of unit vectors in $\mathbb{R}^d$
where all pairwise inner products lie in $\{\alpha, \beta\}$ with $-1 \le \beta
< 0 \le \alpha < 1$. Very recently Jiang, Tidor, Yao, Zhang and Zhao determined
the limit of $N_{\alpha, \beta}(d)/d$ as $d\to\infty$ when $\alpha + 2\beta <
0$ or $(1-\alpha)/(\alpha-\beta) \in \{1,\sqrt2,\sqrt3\}$, and they proposed a
conjecture on the limit in terms of eigenvalue multiplicities of signed graphs.
We establish their conjecture whenever $(1-\alpha)/(\alpha - \beta) <
\lambda^*$.
