Full friendly index sets and full product-cordial index sets of some permutation petersen graphs
Bibliographic record
Abstract
LetG = (V,E) be a connected graph without loops. A vertex labeling g : V [arrow right] Z^sub 2^ induces two edge labelings f^sup +^, f* : E [arrow right] Z^sub 2^, given by f^sup +^(uv) = f(u) + f(v) and f*(uv) = f(u)f(v) for each uv ∈ E respectively. For j ∈ Z^sub 2^, let v^sub f^ (j) = |f^sup -1^(j)|, e^sub f+^(j) = |(f^sup +^)^sup -1^(j)| and e^sub f*^ (j) = |(f*)^sup -1^(j)|. A vertex labeling f is called friendly if |v^sub f^ (1) - v^sub f^ (0)| ≤ 1. For a friendly labeling f of G, the friendly index of G with respect to f is defined to be i^sup +^^sub f^ (G) = e^sup +^^sub f+^(1) - e^sub f+^(0), and the product-cordial index is defined to be i*^sub f^ (G) = e^sub f*^(1) - e^sub f*^(0). The full friendly index set (FFI) and the full product-cordial index set (FPCI) of G contain precisely all the values i^sup +^^sub f^ (G) and i*^sub f^ (G) taken over all friendly labelings of G, respectively. In this paper, we study the FFI and the FPCI of odd twisted cylinder and two permutation Petersen graphs.
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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) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 0.001 |
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".