Subcubic graphs without eigenvalues in $(-1, 1)$

[2601.01482] Subcubic graphs without eigenvalues in $(-1, 1)$ Skip to main content Search Submit Donate Log in Mathematics > Combinatorics arXiv:2601.01482 (math) [Submitted on 4 Jan 2026 ( v1 ), last revised 28 Aug 2026 (this version, v3)] Title: Subcubic graphs without eigenvalues in $(-1, 1)$ Authors: Shenwei Huang , Zilin Jiang View PDF HTML (experimental) Abstract: Guo and Royle recently classified the connected cubic graphs without eigenvalues of their adjacency matrix in the open interval $(-1, 1)$, and raised the question of extending their classification to graphs of maximum degree at most $3$. Together with their cubic classification, our result fully answers this question by characterizing all connected subcubic graphs that are not cubic and have no eigenvalues in $(-1,1)$. We show that exactly two infinite families and seven sporadic examples occur, and that every sporadic graph has at most $18$ vertices. To obtain this complete classification, we build a bridge between spectral graph theory and structural graph theory for graphs whose adjacency matrices, after selected diagonal entries are changed to $-1$, have smallest eigenvalue at least $-2$. This generalizes the classical theorem of Cameron, Goethals, Seidel and Shult for graphs with smallest eigenvalue at least $-2$. As a consequence, we prove that $(-1,1)$ is a maximal spectral gap set for the class of connected subcubic graphs. Guo and Royle, answering a question of Kollár and Sarnak, established this maximality for connected cubic graphs. Our result generalizes their conclusion to the subcubic setting. Comments: 34 pages, 8 figures and 54 diagrams; revised version gives purely structural formulations of the classification theorems and adds ancillary Lean formalization Subjects: Combinatorics (math.CO) MSC classes: 05C50, 05C75, 15A18 Cite as: arXiv:2601.01482 [math.CO] (or arXiv:2601.01482v3 [math.CO] for this version) https://doi.org/10.48550/arXiv.2601.01482 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Zilin Jiang [ view email ] [v1] Sun, 4 Jan 2026 11:07:02 UTC (25 KB) [v2] Wed, 29 Apr 2026 20:46:39 UTC (27 KB) [v3] Fri, 28 Aug 2026 01:34:18 UTC (304 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Ancillary-file links: Ancillary files ( details ) : lean/README.md lean/SubcubicSpectralGap.lean lean/SubcubicSpectralGap/Basic.lean lean/SubcubicSpectralGap/Common/Basic.lean lean/SubcubicSpectralGap/Common/DistanceTwo.lean lean/SubcubicSpectralGap/Common/Figure5.lean lean/SubcubicSpectralGap/Common/Figure5Data.lean 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