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arXiv · 2602.20783

A structure theory for signed graphs with fixed smallest eigenvalue

Meng-Yue Cao·Jack H. Koolen·Jing-Yuan Liu·Qianqian Yang
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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 $\{0,\pm1\}$-vectors of squared norm $2$ and may therefore be viewed as a natural signed generalization of generalized line graphs.

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Meng-Yue Cao, Jack H. Koolen, Jing-Yuan Liu, Qianqian Yang. 2026-02-24. A structure theory for signed graphs with fixed smallest eigenvalue. https://arxiv.org/abs/2602.20783

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