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Explicit nonbipartite Ramanujan graphs of every degree

Theorem 1.1: for every $d\ge3$ there are $n_0(d)$ , $k_d$ and a deterministic algorithm that, on every even $n\ge n_0(d)$ , outputs the adjacency list of a simple connected nonbipartite $d$ -regular graph on $n$ vertices whose nonconstant eigenvalues lie strictly between $\pm2\sqrt{d-1}$ , in $O_d(n^{k_d})$ bit operations. The construction uses partial pairings, deterministic resolvent estimates, cleanup and an exact spectral repair step. It is not strongly explicit (no local adjacency queries), and the exponent may grow with $d$ .

Exact FrontierDelta

Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Explicit nonbipartite Ramanujan graphs of every degree · Explicit Ramanujan graphs, every degree

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Registry verification: unreviewed · announcement · candidate

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VibeMathed record: Explicit nonbipartite Ramanujan graphs of every degree

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No independent mathematician has checked this yet. Checked here: Theorem 1.1 and Section 1.1 were read against the posed problem. The theorem gives, for each fixed $d\ge3$ , a deterministic algorithm that on even $n\ge n_0(d)$ outputs a simple $d$ -regular graph on $n$ vertices in $O_d(n^{k_d})$ bit operations with every nonconstant eigenvalue strictly inside $(-2\sqrt{d-1},2\sqrt{d-1})$ . Scope limits stated by the paper: the exponent and threshold depend on $d$ , running time is not polynomial jointly in $n$ and $d$ , there is no local neighbour-query algorithm, and the Bilu-Linial signing conjecture is not resolved. Existence alone was already known from Huang-McKenzie-Yau. No Lean formalization exists for this family. The proof was not refereed.

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Explicit nonbipartite Ramanujan graphs of every degree

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