A structure theory for signed graphs with fixed smallest eigenvalue

[2602.20783] A structure theory for signed graphs with fixed smallest eigenvalue Skip to main content Search Submit Donate Log in Mathematics > Combinatorics arXiv:2602.20783 (math) [Submitted on 24 Feb 2026 ( v1 ), last revised 27 Aug 2026 (this version, v2)] Title: A structure theory for signed graphs with fixed smallest eigenvalue Authors: Meng-Yue Cao , Jack H. Koolen , Jing-Yuan Liu , Qianqian Yang View PDF HTML (experimental) 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. Subjects: Combinatorics (math.CO) MSC classes: 05C50, 05C22, 52C35, 05C62, 05C75 Cite as: arXiv:2602.20783 [math.CO] (or arXiv:2602.20783v2 [math.CO] for this version) https://doi.org/10.48550/arXiv.2602.20783 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Meng-Yue Cao [ view email ] [v1] Tue, 24 Feb 2026 11:23:18 UTC (56 KB) [v2] Thu, 27 Aug 2026 07:56:03 UTC (57 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.CO < prev | next > new | recent | 2026-02 Change to browse by: math References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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