A structure theory for signed graphs with fixed smallest eigenvalue

Abstract

In this paper, we give a structure theory for signed graphs with fixed smallest eigenvalue. As a consequence, we prove that for every $\lambda \in (-1-\sqrt{2},-2]$, if a connected signed graph has smallest eigenvalue at least $\lambda$ and sufficiently large minimum valency, then its smallest eigenvalue is at least $-2$ and it is $1$-integrable. Thus, every such signed graph admits a representation by a family of $\{0,\pm 1\}$-vectors of squared norm $2$ and may therefore be viewed as a natural signed generalization of generalized line graphs.