D-irregularity Strength of a Graph
Bibliographic record
Abstract
We initiate to study a \(D\)-irregular labeling, which generalizes both non-inclusive and inclusive \(d\)-distance irregular labeling of graphs. Let \(G=(V(G),E(G))\) be a graph, \(D\) a set of distances, and \(k\) a positive integer. A mapping \(\varphi\) from \(V(G)\) to the set of positive integers \(\{1,2,\dots,k\}\) is called a \(D\)-irregular \(k\)-labeling of \(G\) if every two distinct vertices have distinct weights, where the weight of a vertex \(x\) is defined as the sum of labels of vertices whose distance from \(x\) belongs to \(D\). The least integer \(k\) for which \(G\) admits a \(D\)-irregular labeling is the \(D\)-irregularity strength of \(G\) and denoted by \(\mathrm{s}_D(G)\). In this paper, we establish several fundamental properties on \(D\)-irregularity strength for arbitrary graphs. We also determine this parameter exactly for families of graphs with small diameter or small maximum degree.
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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 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".