Some Relations Between Rank, Vertex Cover Number and Energy of Graph
Bibliographic record
Abstract
In this paper, we extend some results of [F. Shaveisi, lower bounds on the vertex cover number and energy of graphs, MATCH Commun. Math. Comput. Chem, 87(3) (2022) 683-692] which state some relations between the vertex cover and other parameters, such as the order and maximum or minimum degree of graphs. Also, we prove that for a graph G, E(G) ≥ 2β(G)−2Ce(G) and so E(G) ≥ 2β(G) − 2C(G), where E(G), β(G), Ce(G) and C(G) denote the energy, vertex cover, number of even cycles and number of cycles in G, respectively. For these both inequalities we investigate their equality. Finally, we give some relations between E(G),γ(G) and γt(G), where γ(G) and γt(G) are domination number and total domination number of G, respectively.
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| 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".