On a Conjecture Concerning the Friendly Index Sets of Trees.
Bibliographic record
Abstract
For a graph G = (V,E) and a binary labeling f : V (G) → Z2, let vf (i) = |f−1(i)|. The labling f is said to be friendly if |vf (1)−vf (0)| ≤ 1. Any vertex labeling f : V (G) → Z2 induces an edge labeling f∗ : E(G) → Z2 defined by f∗(xy) = |f(x)− f(y)|. Let ef (i) = |f∗−1(i)|. The friendly index set of the graph G, denoted by FI(G), is defined by FI(G) = {|ef (1)− ef (0)| : f is a friendly vertex labeling of G }. In [15] Lee and Ng conjectured that the friendly index sets of trees will form an arithmetic progression. This conjecture has been mentioned in [17] and other manuscripts. In this paper we will first determine the friendly index sets of certain caterpillars of diameter four. Then we will disprove the conjecture by presenting an infinite number of trees whose friendly index sets do not form an arithmetic progression.
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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 | 0.003 | 0.019 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.003 | 0.006 |
| Scholarly communication | 0.004 | 0.008 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.003 |
| 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".