Incident vertex pi coloring of middle, total, line and splitting graph for star and double star graph families
Why this work is in the frame
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Bibliographic record
Abstract
<p>Graph Theory was started by Euler after solving the famous Konigsberg bridge problem. The Graph Coloring is among one of the famous topic for research since it has many beautiful theorems on optimization and its applications in numerous fields of science. The Pi coloring is the coloring of graph parts without a recurring pattern. As a result, it is defined as a function from a set of graph elements with similar properties to the power set of colors, so that each set receives a different color set from the power set. In consequence, Incident Vertex Pi coloring of a graph is defined as the coloring of incident vertices for every single edge with Pi coloring. Incident Vertex Pi coloring of the complete graph is <span class="math inline">\(n\)</span>, wheel graph, star graph and double star graph is <span class="math inline">\(n+1\)</span>, diamond, friendship graphs is <span class="math inline">\(\Delta +1\)</span>, and double fan graph is <span class="math inline">\(\Delta +2\)</span>. In this research, we derived the Incident Vertex PI coloring of Star and Double Star graph’s Middle graph, Total graph, Line graph, and Splitting graph.</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 | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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