OpenAI · 178-01 · 2026-09-23 · Combinatorics

Deterministic nonbipartite Ramanujan graphs in every fixed degree

preprints/Deterministic-nonbipartite-Ramanujan-graphs-in-every-fixed-degree-September-23-2026

Family 178. Deterministic nonbipartite Ramanujan graphs in every fixed degree.

The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.

Read against the standing problems

Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N

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

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 $- 2 d - 1$ and $2 d - 1$ .

Preprint source and Lean scope

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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.