On star polynomials of digraphs and their applications to domination
Bibliographic record
Abstract
A directed star is a star digraph in which the centre is either a sink or a source. With every directed star, we associate a weight \(w\left(\alpha\right)\). Let \(D\) be a digraph; and \(C\), a spanning subgraph of \(D\); in which every component is a directed star. Then \(C\) is called a directed star cover of \(D\), and the weight of \(C\) is \(w(C)=\prod_{\alpha}w\left(\alpha\right)\), where the product is taken over all the components \(\alpha\) of \(C\). The directed star polynomial of \(D\) is \[E\left(D;\mathbf{w}\right)=\sum w(C),\] where the sum is taken over all the directed star covers of \(D\). The paper establishes properties of star polynomials and establishes relationships between star polynomials, source polynomials (where all components are source stars), and sink polynomials (where all components are sink stars). Furthermore, it presents methods to compute these polynomials for general digraphs. We derive formulae for the star polynomials of rooted products of digraphs. Moreover, we establish applications of these polynomials to several domination parameters and obtain results regarding these polynomials and independent sets in undirected graphs.
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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 | 0.002 | 0.008 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.006 |
| Science and technology studies | 0.002 | 0.003 |
| Scholarly communication | 0.002 | 0.005 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.003 |
| Insufficient payload (model declined to judge) | 0.012 | 0.002 |
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".