On the metric dimension of Mobius ladders
Bibliographic record
Abstract
If G is a connected graph, the distance d(u, v) between two vertices u, v is an element of V(G) is the length of a shortest path between them. Let W = {w(1), w(2), ... , w(k)} be an ordered set of vertices of G and let v be a vertex of G. The representation r(v vertical bar W) of v with respect to W is the k-tuple (d(v, w(1)), d(v, w(2)), ... , d(v, w(k))). If distinct vertices of G have distinct representations with respect to W, then W is called a resolving set or locating set for G. A resolving set of minimum cardinality is called a basis for G and this cardinality is the metric dimension of G, denoted by dim(G). A family G of connected graphs is a family with constant metric dimension if dim(G) does not depend upon the choice of G in G. In this paper, we are dealing with the study of metric dimension of Mobius ladders. We prove that Mobius ladder M-n constitute a family of cubic graphs with constant metric dimension and only three vertices suffice to resolve all the vertices of Mobius ladder M-n except when n equivalent to 2(mod 8). It is natural to ask for the characterization of regular graphs with constant metric dimension.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.003 | 0.001 |
| Science and technology studies | 0.001 | 0.003 |
| Scholarly communication | 0.002 | 0.003 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".