The metric dimension of circulant graphs and their Cartesian products
Bibliographic record
Abstract
Let G = (V, E) be a connected graph (or hypergraph) and let d(x, y) denote the distance between vertices x, y ∈The minimum cardinality of a resolving set for G is called the metric dimension of G, denoted by β(G).The circulant graph Cn(1, 2, . . ., t) has vertex set {v0, v1, . . ., vn-1} and edges vivi+j where 0 ≤ i ≤ n -1 and 1 ≤ j ≤ t and the indices are taken modulo n (2 ≤ t ≤ n 2 ).In this paper we determine the exact metric dimension of the circulant graphs Cn(1, 2, . . ., t), extending previous results due to Borchert and Gosselin (2013), Grigorious et al. (2014), andVetrík (2016).In particular, we show that β(Cn(1, 2, . . ., t)) = β(Cn+2t(1, 2, . . ., t)) for large enough n, which implies that the metric dimension of these circulants is completely determined by the congruence class of n modulo 2t.We determine the exact value of β(Cn(1, 2, . . ., t)) for n ≡ 2 mod 2t and n ≡ (t + 1) mod 2t and we give better bounds on the metric dimension of these circulants for n ≡ 0 mod 2t and n ≡ 1 mod 2t.In addition, we bound the metric dimension of Cartesian products of circulant graphs.
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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) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".