Bibliographic record
Abstract
Abstract A blow-up of n copies of a graph G is the graph obtained by replacing every vertex of G by an independent set of size n , where the copies of two vertices in G are adjacent in the blow-up if and only if they are adjacent in G . In this work, we characterize strong cospectrality, periodicity, perfect state transfer (PST) and pretty good state transfer (PGST) in blow-up graphs. We prove that if a blow-up admits PST or PGST, then n = 2. In particular, if G has an invertible adjacency matrix, then each vertex in a blow of two copies of G pairs up with a unique vertex to exhibit strong cospectrality. Under mild conditions, we show that periodicity (resp., almost periodicity) of a vertex in G guarantees PST (resp. PGST) between the two copies of the vertex in the blow-up. This allows us to construct new families of graphs with PST from graphs that do not admit PST. We also characterize PST and PGST in the blow-ups of complete graphs, paths, cycles and cones. Finally, while trees in general do not admit PST, we provide infinite families of stars and subdivided stars whose blow-ups admit PST.
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How this classification was reachedexpand
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 itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, 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".