The partition dimension of corona product graphs
Bibliographic record
Abstract
Given a set of vertices S = {ν1,ν2, -,νk} of a connected graph G, the metric representation of a vertex ν of G with respect to S is the vector r(ν|5) = (d(ν, ν1),d(ν, ν2),···,d(ν, νk)), where d(ν,νi), i ∈ {1,···,k} denotes the distance between ν and νi. S is a resolving set of G if for every pair of distinct vertices u, ν of G, r(u|S) ¢ r (ν|S). The metric dimension dim(G) of G is the minimum cardinality of any resolving set of G. Given an ordered partition II = {P1,P2,···,Pt} of vertices of a connected graph G, the partition representation of a vertex ν of G, with respect to the partition II is the vector r(ν|II) = (d(ν,P1),d(ν,P2),···,d(ν,Pt)), where d(ν,Pi), 1 ≤ i ≤ t, represents the distance between the vertex ν and the set Pi, that is d(ν, Pi) = minu∈pi{d(ν,u)}. II is a resolving partition for G if for every pair of distinct vertices u, ν of G, r(u|II) ¢ r(ν|II). The partition dimension pd(G) of G is the minimum number of sets in any resolving partition for G. Let G and H be two graphs of order n1 and n2 respectively. The corona product G o H is defined as the graph obtained from G and H by taking one copy of G and n1 copies of H and then joining by an edge, all the vertices from the ith-copy of H with the ith-vertex of G. Here we study the relationship between pd(G o H) and several parameters of the graphs G o H, G and H, including dim(G o H), pd(G) and pd(H).
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.004 | 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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".