The Gutman-Milovanović index and some Hamiltonian properties of a graph
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Bibliographic record
Abstract
<p>Let <span class="math inline">\(G = (V, E)\)</span> be a graph. The Gutman-Milovanović index of a graph <span class="math inline">\(G\)</span> is defined as <span class="math inline">\(\sum\limits_{uv \in E} (d(u) d(v))^{\alpha}(d(u) + d(v))^{\beta}\)</span>, where <span class="math inline">\(\alpha\)</span> and <span class="math inline">\(\beta\)</span> are any real numbers and <span class="math inline">\(d(u)\)</span> and <span class="math inline">\(d(v)\)</span> are the degrees of vertices <span class="math inline">\(u\)</span> and <span class="math inline">\(v\)</span> in <span class="math inline">\(G\)</span>, respectively. In this note, we present sufficient conditions based on the Gutman-Milovanović index with <span class="math inline">\(\alpha > 0\)</span> and <span class="math inline">\(\beta >0\)</span> for some Hamiltonian properties of a graph. We also present upper bounds for the Gutman-Milovanović index of a graph for different ranges of <span class="math inline">\(\alpha\)</span> and <span class="math inline">\(\beta\)</span>.</p>
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.001 | 0.002 |
| 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