On the Cordial of Weak Labeling of an l-fold Cycle Graph, T_l(C_n)
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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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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.020 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.003 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.003 | 0.001 |
| Research integrity | 0.000 | 0.002 |
| Insufficient payload (model declined to judge) | 0.000 | 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 it