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A proof of a conjecture of Cazenave on graph eigenvalues

Primarily AI-generated textHuman understanding: some partsmath.CO — Combinatorics

Contributed by Alexander Kramer ↗

Version 1 / Nov 10, 2026 / CC0 1.0

Abstract

We prove Cazenave's conjecture μ22≥4λ2+1\mu_2^2\ge4\lambda_2+1 for connected graphs, where λ2\lambda_2 and μ2\mu_2 are the second largest adjacency and Laplacian eigenvalues. We also characterize equality in this inequality and in the linear bound μ2≥λ2+1\mu_2\ge\lambda_2+1.

Provenance statement

The proof and text were developed with AI assistance, including OpenAI Codex, for proof generation, refinement, literature review, and computational checks. The author reviewed the arguments and found no errors, but did not independently work out every step.

Tools used

OpenAI
Codex

References

  1. A. E. Brouwer and W. H. Haemers, Spectra of Graphs , Universitext, Springer, New York, 2012. doi:10.1007/978-1-4614-1939-6 (https://doi.org/10.1007/978-1-4614-1939-6).DOI
  2. T. Cazenave, Lemmas and Conjectures in Spectral Graph Theory , arXiv:2611.03879v1 [math.CO], 2 November 2026. doi:10.48550/arXiv.2611.03879 (https://arxiv.org/abs/2611.03879).arXivDOI
  3. J. Li, J.-M. Guo and W. C. Shiu, On the second largest Laplacian eigenvalues of graphs, Linear Algebra Appl. 438 (2013), 2438--2446. doi:10.1016/j.laa.2012.10.047 (https://doi.org/10.1016/j.laa.2012.10.047).DOI

Version history

  1. v1 · 2 pagesSubmitted by Alexander KramerInitial depositCurrentNov 09, 2026