Summary: Let $G$ be a connected graph with least eigenvalue -2, of multiplicity $k$. A star complement for -2 in $G$ is an induced subgraph $H = G - X$ such that $| X| = k$ and -2 is not an eigenvalue of $H$. In the case that $G$ is a generalized line graph, a characterization of such subgraphs is used to decribe the eigenspace of -2. In some instances, $G$ itself can be characterized by a star complement. If $G$ is not a generalized line graph, $G$ is an $exceptional$ graph, and in this case it is shown how a star complement can be used to construct $G$ without recourse to root systems.

graph, eigenvalue, eigenspace