The partition dimension of corona product graphs
Why this work is in the frame
A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.
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).
Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.
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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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