On the Cordial of Weak Labeling of an l-fold Cycle Graph, T_l(C_n)
Bibliographic record
Abstract
We investigate a number-theoretic graph labeling known as k-prime cordial labeling, where each vertex of a graph G is assigned a label from the set {1,2, ... , k}, and each edge receives a label equal to the greatest common divisor of its endpoint labels. A weak labeling is k-prime cordial if the number of vertices labeled with each integer differs by at most two, and the number of edges labeled 1 differs from those not labeled 1 by at most two. In this paper, we introduce the concept of the $\ell$-fold of a graph, denoted T_l(G), constructed by joining corresponding vertices across l copies of a base graph G. We focus on the case where G is a cycle graph C_n and show that T_l(C_n) admits a 4-prime cordial weak labeling for all l≥2. This result extends previous work on trigraphs and contributes to the broader understanding of cordial labeling in replicated graph structures.
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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 | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 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".