A strengthening and a multipartite generalization of the Alon-Boppana-Serre theorem
Bibliographic record
Abstract
The Alon-Boppana theorem confirms that for every ε > 0 \varepsilon >0 and every integer d ≥ 3 d\ge 3 , there are only finitely many d d -regular graphs whose second largest eigenvalue is at most 2 d − 1 − ε 2\sqrt {d-1}-\varepsilon . Serre gave a strengthening showing that a positive proportion of eigenvalues of any d d -regular graph must be bigger than 2 d − 1 − ε 2\sqrt {d-1}-\varepsilon . We provide a multipartite version of this result. Our proofs are elementary and also work in the case when graphs are not regular. In the simplest, monopartite case, our result extends the Alon-Boppana-Serre result to non-regular graphs of minimum degree d d and bounded maximum degree. The two-partite result shows that for every ε > 0 \varepsilon >0 and any positive integers d 1 , d 2 , d d_1,d_2,d , every n n -vertex graph of maximum degree at most d d , whose vertex set is the union of (not necessarily disjoint) subsets V 1 , V 2 V_1,V_2 , such that every vertex in
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.009 | 0.029 |
| Meta-epidemiology (narrow) | 0.002 | 0.002 |
| Meta-epidemiology (broad) | 0.003 | 0.008 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.006 | 0.009 |
| Scholarly communication | 0.006 | 0.024 |
| Open science | 0.004 | 0.012 |
| Research integrity | 0.004 | 0.010 |
| Insufficient payload (model declined to judge) | 0.039 | 0.013 |
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.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".