Sharp Bounds for Spectral Radius of Graphs Presented by K-Neighbour of the Vertices
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Bibliographic record
Abstract
Let G = (V, E) be a simple connected graph with V (G) = {v1, v2, …, vn} and degree sequence d1, d2, …, dn. Denote , where k is a positive integer number and vi∈V(G) and note that t0(i)=di. Let ρ(G) be the largest eigenvalue of adjacent matrix of G. In this paper, we present sharp upper and lower bounds of ρ(G) in terms of mk(i) (see theorem (2.1)). From which, we can obtain some known results, and our result is better than other results in some case.
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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 | 0.002 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.007 |
| Science and technology studies | 0.001 | 0.006 |
| Scholarly communication | 0.000 | 0.010 |
| Open science | 0.007 | 0.001 |
| 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