Result 178, Combinatorics

Deterministic nonbipartite Ramanujan graphs in every fixed degree

For every fixed d ≥ 3, constructs a simple d-regular nonbipartite Ramanujan graph on every sufficiently large even number n of vertices, with every nonconstant adjacency eigenvalue strictly between −2d−1-2\sqrt{d-1} and 2d−12\sqrt{d-1}. A deterministic algorithm outputs the full adjacency list in polynomial bit time, with exponent depending on d.

Algorithm or complexity result

The bigger picture

Why it matters

A graph can give each vertex only a few neighbors while still letting random walks spread efficiently throughout it. The manuscript claims a deterministic construction of such graphs with precise guarantees on this behavior and flexible even sizes.

What changes?

The manuscript reports simple graphs, with no loops or repeated edges, for each fixed degree d at least 3 and every sufficiently large even size n. Every vertex has d neighbors, and the graph is nonbipartite: no two-group split makes every edge cross groups. Every nonconstant adjacency eigenvalue lies strictly between minus and plus twice the square root of (d minus one). A deterministic algorithm outputs the full adjacency list in polynomial bit time, with exponent possibly depending on d.

What does that help mathematicians do?

Adjacency eigenvalues describe how a graph redistributes information between neighboring vertices. The reported bounds imply connectivity and control how quickly random walks approach a uniform distribution, excluding persistent alternation between two groups. Researchers would therefore have deterministic access to graphs with strong spectral expansion at every sufficiently large even size for each fixed degree, rather than merely an unspecified sequence of sizes.

Are there practical applications?

Its immediate value is foundational: it supplies explicitly constructible sparse graphs for studying expansion and mixing, with degree and eventual even size prescribed. Polynomial bit complexity is a theoretical efficiency guarantee, not evidence of practical speed. The degree-dependent exponent and unspecified size threshold leave implementation costs unresolved.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Deterministic nonbipartite Ramanujan graphs in every fixed degree

September 23, 2026 85 pages

For every fixed integer d ≥ 3, we give a deterministic algorithm that constructs a simple nonbipartite d-regular Ramanujan graph on every sufficiently large even number n of vertices. It outputs the full adjacency list in polynomially many bit operations, with an exponent that may depend on d. Every nonconstant adjacency eigenvalue lies strictly between −2d−1-2\sqrt{d-1} and 2d−12\sqrt{d-1}.

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Cite (BibTeX) 
@misc{OAI:Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026,
  author = {{OpenAI}},
  title = {{Deterministic nonbipartite Ramanujan graphs in every fixed degree}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026/paper.pdf}{OAI:Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026}},
  year = {2026}
}

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Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

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