Jing-Yuan Liu | Searcharxiv

arXiv2026

A structure theory for signed graphs with fixed smallest eigenvalue

In this paper, we give a structure theory for signed graphs with fixed smallest eigenvalue. As a consequence, we prove that for every $λ\in(-1-\sqrt{2},-2]$, if a connected signed graph has smallest eigenvalue at least $λ$ 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 $\{0,\pm1\}$-vectors of squared norm $2$ and may therefore be viewed as a natural signed generalization of generalized line graphs.

math.CO↗

arXiv2026

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

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.

math.CO↗