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The smallest eigenvalue of a graph is the smallest eigenvalue of itsadjacency matrix. We show that the family of graphs with smallest eigenvalue atleast $-\lambda$ can be defined by a finite set of forbidden induced subgraphsif and only if $\lambda < \lambda^*$, where $\lambda^* = \beta^{1/2} +\beta^{-1/2} \approx 2.01980$ and $\beta$ 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, supplementingHoffman's work on those limit points in $[-2, \infty)$.  We also prove that the same conclusion about forbidden subgraphcharacterization holds for signed graphs, and we give an application tosymmetric integer matrices whose diagonal entries are all zero. Our impetus forthe study of signed graphs is to determine the maximum cardinality of aspherical two-distance set with two fixed angles (one acute and one obtuse) inhigh dimensions. Denote by $N_{\alpha, \beta}(n)$ the maximum number of unitvectors 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 ofeigenvalue multiplicities of signed graphs. We establish their conjecturewhenever $(1-\alpha)/(\alpha - \beta) < \lambda^*$.

Forbidden induced subgraphs for graphs and signed graphs with eigenvalues bounded from below