Let $G$ be an undirected simple graph and let $A$ be its adjacency matrix. It is easy to see that $A$ is neither positive semidefinite nor negative semidefinite.
I would like to know if there are interesting graphs $G$ for which smallest eigenvalue of $A$ is at least $-1$. Clearly, cliques and disjoint union of cliques have this property. Are there any other graphs with this property?
Slightly related: A quick google search reveals that graphs with smallest eigenvalue $-2$ have been studied. What is special about $-2$ (or not special about $-1$)?
A graph with least eigenvalue at least $-1$ is a disjoint union of cliques. (Proof: the least eigenvalue of $K_{1,2}$ is $-\sqrt2$, interlacing.) The graphs with least eigenvalue at least $-2$ were characterized by Cameron, Goethals and Seidel. They are line graphs, so-called generalized line graphs, and a finite set of graphs associated to $E_6$, $E_7$, $E_8$ root systems.