Bibliographic record
Abstract
Let G be a graph with n vertices and μ1, μ2, ..., μn be the Laplacian eigenvalues of G. The Laplacian-energy-like graph invariant LEL(G) = ∑n i=1 √ μi, has been defined and investigated in [1]. Two non-isomorphic graphs G1 and G2 of the same order are said to be LEL-equienergetic if LEL(G1) = LEL(G2). In [2], three pairs of LELequienergetic non-cospectral connected graphs are given. It is also claimed that the LEL-equienergetic non-cospectral connected graphs are relatively rare. It is natural to consider the question: Whether the number of the LEL-equienergetic non-cospectral connected graphs is finite? The answer is negative, because we shall construct a pair of LEL-equienergetic non-cospectral connected graphs of order n, for all n ≥ 12 in this paper.
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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 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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 | 0.001 |
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".