Odd harmonious labeling on squid graph and double squid graph
Bibliographic record
Abstract
Abstract An injective functionffrom set of vertices in graphGto a set of {0,1,…,|E| − 1} is called an odd harmonious labeling if the functionfinduced the edge functionf* from the set of edges ofGto a set of odd positive integer number {1,3,5,…,2|E| − 1} withf*(xy) =f(x) +f(y) for every edgexyinE.Graph that has an odd harmonious labeling is called odd harmonious graph. The squid graphTn,kis a graph which is obtained from a cycleCnand we addkpendant to one vertex of the cycle. It is known thatCnis an odd harmonious graph if and only ifn= 0 mod 4. However, by adding at least one pendant in the cycle graph, we can label the new graph odd harmoniously for all even number of vertices. In this paper, we showed that the graphTn,kandT2n,kare an odd harmonious graph, forn= 0 (mod 2),n≥ 4 andk≥ 1. The construction of the odd harmonious labeling of the graphTn,kandT2n,kare inspired by the odd harmonious labeling ofCnforn= 0(mod 4).
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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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".