Properties of characteristic polynomial and eigenvalues of antiadjacency matrix of directed unicyclic helm graph
Bibliographic record
Abstract
Abstract A directed unicyclic graph is a directed graph that has only one directed cycle subgraph. A directed unicyclic helm graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">→</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> is obtained from a directed wheel graph <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <mml:mrow> <mml:mover accent="true"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>W</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:mrow> <mml:mo stretchy="true">→</mml:mo> </mml:mrow> </mml:mover> </mml:mrow> </mml:math> by adjoining a directed pendant edge at each vertex of the cycle. A directed graph can be represented into several matrix representations, one of them is the antiadjacency matrix. The antiadjacency matrix is a matrix in which the entries represent whether there is a directed edge from one vertex to another. This paper discusses the general form of the coefficients of the characteristic polynomial that obtained by adding all of the determinants of antiadjacency matrix from each induced acyclic and cyclic subgraphs. The eigenvalues of the antiadjacency matrix of the directed unicyclic helm graph obtained by polynomial factorization. The result obtained denotes that the coefficients of the characteristic polynomial and eigenvalues of the antiadjacency matrix depend on the number of vertices of the cycle subgraphs of directed unicyclic helm graph.
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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) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| 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".