Forward Difference Properties of the (n, k)-Star Graph and Some Other Interconnection Networks
Bibliographic record
Abstract
An important invariant of an interconnection network is its surface area, the number of vertices at distance i from a node. Although much work has been done to obtain formulas for the surface areas for many interconnection networks, most of the formulas are not in the so-called closed form except for a very few trivial graphs. It is known that for an interconnection network, if its surface area satisfies the so-called forward difference property, then for any specific distance i, its surface area of radius i in closed form (a polynomial of degree i) can be obtained, provided that we have i + 1 initial values of the surface area of radius i. This property is known to hold for the hypercube and the star graph. We show in this paper that the property also holds for the (n, k)-star graph, 1 ≤ k ≤ n − 1, a family of interconnection networks that also include the star graph when k = n − 1. We then show that the technique we use for the result is general that can also be used to prove the property for some other networks.
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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 | 0.000 | 0.002 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.001 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 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".