Metric Dimension of Corona Product and Join Graph of Zero Divisor Graphs of Direct Product of Finite Fields
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Bibliographic record
Abstract
The metric dimension of a graph is the smallest number of vertices such that all vertices are uniquely determined by their distances to the chosen vertices. The corona product of graphs G and H is the graph G ⊙ H obtained by taking one copy of G , called the center graph, | V ( G ) | copies of H , called the outer graph, and making the j t h vertex of G adjacent to every vertex of the j t h copy of H , where 1 ⩽ j ⩽ | V ( G ) | . The Join graph G + H of two graphs G and H is the graph with vertex set V ( G + H ) = V ( G ) ∪ V ( H ) and edge set E ( G + H ) = E ( G ) ∪ E ( H ) ∪ { u v : u ∈ V ( G ) , v ∈ V ( H ) } . In this paper, we determine the Metric dimension of Corona product and Join graph of zero divisor graphs of direct product of finite fields.
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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 | 0.001 | 0.002 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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