The metric dimension of circulant graphs and their Cartesian products
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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 V (G). A subset W V (G) is called a resolving set for G if for every pair of distinct vertices x, y V (G), there is w W such that d(x, w) = d(y, w). The minimum cardinality of a resolving set for G is called the metric dimension of G, denoted by (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), In particular, we show that (Cn(1, 2, . . . , t)) = (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 (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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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 | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 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