Degree sequence of the generalized Sierpiński graph
Bibliographic record
Abstract
Sierpiński graphs are studied in fractal theory and have applications in diverse areas including dynamic systems, chemistry, psychology, probability, and computer science. Polymer networks and WK-recursive networks can be modeled by generalized Sierpiński graphs. The degree sequence of (ordinary) Sierpi\'nski graphs and Hanoi graphs (and some of their topological indices) are determined in the literature. The number of leaves (vertices of degree one) of the generalized Sierpiński graph $S(T, t)$ of any tree $T$ was determined in 2017 and in terms of $t$, $|V(T)|$, and the number of leaves of the base graph $T$. In this paper, we generalize these results. More precisely, for every simple graph $G$ of order $n$, we completely determine the degree sequence of the generalized Sierpiński graph $S(G,t)$ of $G$ in terms of $n$, $t$ and the degree sequence of $G$. By using it, we determine the exact value of the general first Zagreb index of $S(G,t)$ in terms of the same parameters of $G$.
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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 | 0.002 |
| 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 | 0.001 | 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".