Tight upper bounds for the p-anionic Clar number of fullerenes
Bibliographic record
Abstract
A fullerene is an all-carbon molecule with a polyhedral structure whereeach atom is bonded to three other atoms and each face is either apentagon or a hexagon. Fullerenes correspond to 3-regular planar graphswhose faces have sizes 5 or 6. The p-anionic Clar number C_(p)(G) of afullerene G is equal to p + h, where h is maximized over all choices ofp + h independent faces (exactly p pentagons and h hexagons) thedeletion of whose vertices leave a graph with a perfect matching. Thisdefinition is motivated by the chemical observation that pentagonalrings can accommodate an extra electron, so that the pentagons of afullerene with charge −p, compete with the hexagons to host ‘Clarsextets’ of six electrons, and pentagons will preferentially acquire thep excess electrons of the anion. Tight upper bounds are established for the p-anionic Clar number offullerenes for p > 0. The upper bounds are derived via graph theoreticarguments and new results on minimal cyclic-k-edge cutsets in IPRfullerenes (fullerenes that have all pentagons pairwise disjoint). Thesebounds are shown to be tight by infinite families of fullerenes thatachieve them.
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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 | 0.003 | 0.025 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.005 | 0.004 |
| Science and technology studies | 0.004 | 0.005 |
| Scholarly communication | 0.007 | 0.014 |
| Open science | 0.004 | 0.007 |
| Research integrity | 0.003 | 0.007 |
| Insufficient payload (model declined to judge) | 0.015 | 0.003 |
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".