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Energy of a set of vertices in a Graph

2007· article· en· W2189028915 on OpenAlex

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A frame that forgets how it found something cannot be audited. These are the routes that admitted this work.

affAt least one author lists a Canadian institution in the pinned OpenAlex snapshot.

Bibliographic record

VenueAKCE International Journal of Graphs and Combinatorics · 2007
Typearticle
Languageen
FieldMathematics
TopicGraph theory and applications
Canadian institutionsWomen's College Hospital
Fundersnot available
KeywordsMathematicsCombinatoricsVertex (graph theory)Incidence matrixEigenvalues and eigenvectorsDiagonalGraph energyGraphDiagonal matrixDiscrete mathematicsGraph powerPhysicsGeometryLine graphQuantum mechanics

Abstract

fetched live from OpenAlex

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.

Fetched live from OpenAlex and de-inverted. Abstracts are not stored in this database: the inverted indexes are 8.6 GB of the frame’s 9.3 GB of text, and the host has 13 GB free.

Full frame distilled prediction

Teacher imitation

Not 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.

metaresearch head score (Codex)0.001
metaresearch head score (Gemma)0.000
Version: codex-gemma-dda1882f352aValidation status: machine_predicted_unvalidated
Candidate categoriesnone
Consensus categoriesnone
DomainCandidate signal: none · Consensus signal: none
Study designCandidate signal: Theoretical or conceptual · Consensus signal: Theoretical or conceptual
GenreCandidate signal: Empirical · Consensus signal: Empirical
Teacher disagreement score0.011
Threshold uncertainty score0.251

Codex and Gemma teacher scores by category

CategoryCodexGemma
Metaresearch0.0010.000
Meta-epidemiology (narrow)0.0000.000
Meta-epidemiology (broad)0.0000.000
Bibliometrics0.0000.000
Science and technology studies0.0000.000
Scholarly communication0.0000.000
Open science0.0000.000
Research integrity0.0000.000
Insufficient payload (model declined to judge)0.0000.000

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.

Opus teacher head0.023
GPT teacher head0.311
Teacher spread0.289 · how far apart the two teachers sit on this one work
Validation statusscore_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it