Further Results on Radio Number of Wedge sum of Graphs
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
Bibliographic record
Abstract
Let G be a simple connected graph with vertex set V and diameter d . An injective function c : V → { 1 , 2 , 3 , … } is called a radio labeling of G if | c ( x ) c ( y ) | + d ( x , y ) ≥ d + 1 for all distinct x , y ∈ V , where d ( x , y ) is the distance between vertices x and y . The largest number in the range of c is called the span of the labeling c . The radio number of G is the minimum span taken over all radio labelings of G . For a fixed vertex z of G , the sequence ( l 1 , l 2 , … , l r ) is called the level tuple of G , where l i is the number of vertices whose distance from z is i . Let J k ( l 1 , l 2 , … , l r ) be the wedge sum (i.e. one vertex union) of k ≥ 2 graphs having same level tuple ( l 1 , l 2 , … , l r ) . Let J ( l 1 l ′ 1 , l 2 l ′ 2 , … , l r l ′ r ) be the wedge sum of two graphs of same order, having level tuples ( l 1 , l 2 , … , l r ) and ( l ′ 1 , l ′ 2 , … , l ′ r ) . In this paper, we compute the radio number for some sub-families of J k ( l 1 , l 2 , … , l r ) and J ( l 1 l ′ 1 , l 2 l ′ 2 , … , l r l ′ r ) .
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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 | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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