Bibliographic record
Abstract
Abstract. A subsetWof the vertex set of a graphGis called aresolving setofGif for every pair of distinct verticesu,vofG, there isw∊Wsuch that the distance ofwanduis different from the distance ofwandv. The cardinality of a smallest resolving set is called the metric dimension ofG, denoted by dim(G). The circulant graphCn(1, 2, . . . ,t) consists of the verticesv0,v1, . . . ,vn−1and the edgesvivi+j, where 0 ≤i≤n− 1, 1 ≤j≤t( ), the indices are taken modulon. Grigorious, Manuel, Miller, Rajan, and Stephen proved that dim(Cn(1, 2, . . . ,t)) ≥t+ 1 for , and they presented a conjecture saying that dim(Cn(1, 2, . . . ,t)) =t+p− 1 forn= 2tk+t+p, where 3 ≤p≤t+ 1. We disprove both statements. We show that ift≥ 4 is even, there exists an infinite set of values ofnsuch that dim(Cn(1, 2, . . . ,t)) =t. We also prove that dim(Cn(1, 2, . . . ,t)) ≤t+p/2 forn= 2tk+t+p, wheretandpare even,t≥ 4, 2 ≤p≤t, andk≥ 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.
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 | 0.000 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 0.000 |
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.
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".