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Record W2051189409 · doi:10.1515/forum.2008.035

Expander graphs and gaps between primes

2008· article· en· W2051189409 on OpenAlex

Why this work is in the frame

A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueForum Mathematicum · 2008
Typearticle
Languageen
FieldMathematics
TopicGraph theory and applications
Canadian institutionsQueen's University
Fundersnot available
KeywordsRamanujan's sumMathematicsExpander graphCombinatoricsPrime powerDiscrete mathematicsChordal graphPrime (order theory)Graph

Abstract

fetched live from OpenAlex

Abstract. The explicit construction of infinite families of d-regular graphs which are Ramanujan is known only in the case d 1 is a prime power. In this paper, we consider the case when d 1 is not a prime power. The main result is that by perturbing known Ramanujan graphs and using results about gaps between consecutive primes, we are able to construct infinite families of ‘‘almost’ ’ Ramanujan graphs for almost every value of d. More precisely, for any fixed e> 0 and for almost every value of d (in the sense of natural density), there are infinitely many d-regular graphs such that all the non-trivial pffiffiffiffiffiffiffiffiffiffiffi eigenvalues of the adjacency matrices of these graphs have absolute value less than ð2 þ eÞ d 1.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.

metaresearch head score (Codex)0.000
metaresearch head score (Gemma)0.000
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesnone
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: Empirical
Teacher disagreement score0.012
Threshold uncertainty score0.596

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0000.000
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0000.000

Machine scores (provisional)

The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.

Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.

Opus teacher head0.050
GPT teacher head0.301
Teacher spread0.250 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it