Energy of a set of vertices in a Graph
Bibliographic record
Abstract
Given a finite graph G = (V, E), and any proper subset D of the vertex set V:= V(G) of G, we associate a nonnegative integral matrix AD(G) = (aij) of order |D| x |D| with D so that the ith diagonal entry in the matrix counts precisely the number of edges that join the ith vertex of D with vertices in V - D so that these partial degrees of the vertices in D are precisely the eigenvalues of AD (G) whence their sum may be conceived as the energy ϵD (G) of the given set D. Invoking the underlying notion of incidence matrix of D, we introduce in this paper the notion of robust domination energy (or, rd-energy) and shear domination energy (or, sd-energy) of G as the maximum (minimum, respectively) energy of a minimal dominating set in G. We raise several interesting open problems and connections of these notions with other well known ones in graph theory.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 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 source (direct Gemma or distilled Codex), 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".