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 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) | 0.000 | 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".