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Ultra-Sparse Expanders and the Free Method

Authors Gil Cohen , Gal Maor

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LIPIcs.APPROX-RANDOM.2026.49.pdf
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ACM Subject Classification
  • Mathematics of computing → Spectra of graphs
  • Theory of computation → Expander graphs and randomness extractors
Keywords
  • Spectral expanders
  • free probability

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Abstract

In this paper we ask how much expansion one can retain with almost no edges beyond connectivity. Concretely, for graphs of average degree 2+ε, what is the "Ramanujan bound" - how does spectral expansion scale with ε? We compare five ultra–sparse graph models - including the configuration model, subdivision of regular expanders, and the union of a cycle with a partial matching - and analyze each under the normalized or unnormalized notions of expansion. In the normalized setting, we prove bounds that are essentially optimal, determining the correct asymptotic dependence on ε. Our results extend to expansion in general irregular graphs.
For some models we prove rigorous bounds - primarily via finite free probability - while others remain beyond our current techniques. To bridge this gap, we introduce the Free Method, which produces quantitative predictions without proving existence - analogous to the probabilistic method, which certifies existence without providing an explicit construction. These predictions align with experiments. We expect the free method to be useful more broadly in graph-theoretic settings.

Cite As Get BibTex

Gil Cohen and Gal Maor. Ultra-Sparse Expanders and the Free Method. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 392, pp. 49:1-49:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2026.49

Author Details

Gil Cohen
  • Tel Aviv University, Israel
Gal Maor
  • Tel Aviv University, Israel

Funding

Funded by ERC starting grant 949499 and by the Israel Science Foundation grant 2989/24.

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