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 $G$.
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.
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 | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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".