Minimum strong diameter of the strong product of complete multipartite graph and path
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Bibliographic record
Abstract
<p>Suppose <span class="math inline">\(G_1=(V_1, E_1)\)</span> is a graph and <span class="math inline">\(G_2=(V_2, E_2)\)</span> is a strong digraph of <span class="math inline">\(G_1\)</span>, where <span class="math inline">\(V_1\)</span> and <span class="math inline">\(V_2\)</span> represent the vertex sets, <span class="math inline">\(E_1\)</span> and <span class="math inline">\(E_2\)</span> represent the edge sets. Let <span class="math inline">\(u\)</span> and <span class="math inline">\(v\)</span> be any two vertices of <span class="math inline">\(G_2\)</span>. The strong distance <span class="math inline">\(sd(u,v)\)</span> is the minimum value of edges in a strong subdiagraph of <span class="math inline">\(G_2\)</span> that contains <span class="math inline">\(u\)</span> and <span class="math inline">\(v\)</span>. The minimum strong diameter of <span class="math inline">\(G_2\)</span> is defined as the maximum eccentricity <span class="math inline">\(se(u)\)</span> from <span class="math inline">\(u\)</span> to all other vertices in <span class="math inline">\(G_2\)</span>. In this paper, we propose different strong orientation methods to explore the minimum strong diameter of the strong product graph of <span class="math inline">\(K_{m_1,m_2,\ldots,m_k}\otimes P_n\)</span>, where <span class="math inline">\(K_{m_1,m_2,\ldots,m_k}\)</span> and <span class="math inline">\(P_n\)</span> represent respectively complete multipartite graph and path. In addition, based on strong orientation methods, a new algorithm is proposed to model the presence or absence of a minimum strong diameter in a strong product graph. Simulation experiments show a trend of simultaneous decrease and concentration in the minimum strong diameter of the strong product graph, as the value of parts in <span class="math inline">\(K_{m_1,m_2,\ldots,m_k}\)</span> increases while the length of <span class="math inline">\(P_n\)</span> remains constant.</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 | 0.001 | 0.001 |
| 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 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.001 |
| 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