Group distance magic set of group vertex magic graphs
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Bibliographic record
Abstract
<p>Let <span class="math inline">\(G\)</span> be a graph of order <span class="math inline">\(n\)</span> and let <span class="math inline">\(A\)</span> be an additive Abelian group with identity 0. A mapping <span class="math inline">\(l : V(G) \to A \setminus \{0\}\)</span> is said to be a <span class="math inline">\(A\)</span>-vertex magic labeling of <span class="math inline">\(G\)</span> if there exists a <span class="math inline">\(\mu \in\)</span> <span class="math inline">\(A\)</span> such that <span class="math inline">\(w(v) = \sum\limits_{u \in N_G(v)} l(u) = \mu\)</span> for all <span class="math inline">\(v \in V\)</span> and <span class="math inline">\(\mu\)</span> is called a magic constant of <span class="math inline">\(\ell\)</span>. The group distance magic set of an <span class="math inline">\(A\)</span>-vertex magic graph <span class="math inline">\(gdms(G,A)\)</span> is defined as <span class="math inline">\(gdms(G,A):= \{ \lambda: \lambda \text{ is a magic constant of some $A$-vertex magic labeling} \}\)</span>. In this paper, we investigate under what conditions <span class="math inline">\(gdms(G,A)\)</span> is a subgroup of <span class="math inline">\(A\)</span>. We also introduce the concept of the reduced group distance magic set, <span class="math inline">\(rgdms(G, A)\)</span>, which can be used as a tool to determine <span class="math inline">\(gdms(G, A)\)</span>.</p>
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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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 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 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| 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