Pell Labeling in Special Graph Classes: An Exploration of Cycles, Stars, and Related Structures
Bibliographic record
Abstract
In a graph 𝐺, define Pell labeling is a map 𝑓: 𝑉(𝐺) → {0,1, ⋯ , 𝑝 -1} with an induced function 𝑓 * : 𝐸(𝐺) → 𝑁 defined by 𝑓 * (𝑢𝑣) = 𝑓(𝑢) + 2𝑓(𝑣) for every 𝑢𝑣 ∈ 𝐸(𝐺) are all distinct where 𝑢, 𝑣 ≤ 0. In addition to this a graph which admits Pell labeling concept is known as Pell graph.In this paper, the Pell labeling concepts applied for the following graphs such as splitting of Star, cycle with parallel chords, alternate double triangular Snake, ladder, Triangular ladder, diagonal ladder, shadow and splitting of path, bistar, subdivision of bistar, prism, 𝐵 𝑛,𝑛 2 and friendship graphs are studied.Our analysis contributes to the understanding of Pell labeling across a broad spectrum of graph configurations, highlighting its applicability and the unique characteristics of each considered graph family.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.004 |
| Open science | 0.001 | 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".