Signed graphs with fixed smallest eigenvalue at least $-3$ and their lattices

Abstract

In this paper, we consider connected signed graphs with smallest eigenvalue at least $-3-\varepsilon$ for a small positive constant $\varepsilon$. We prove that if such a signed graph has sufficiently large minimum valency, then its smallest eigenvalue is at least $-3$, and the lattice associated with it, which is generated by squared norm $3$ vectors, is a sublattice of a direct sum of the standard lattice $\mathbb{Z}^n$ and copies of the root lattice $E_8$. Moreover, there exist infinitely many connected signed graphs with smallest eigenvalue at least $-3$ containing it as a proper induced subgraph. Furthermore, we discuss signed graphs with smallest eigenvalue $-3$ arising from rootless irreducible unimodular lattices.