The generalized connectivity of complete bipartite graphs
Bibliographic record
Abstract
Let G be a nontrivial connected graph of order n, and k an integer with 2 � kn. For a set S of k vertices of G, let �(S) denote the maximum numberof edge-disjoint trees T1;T2;:::;Tin G such that V (Ti) V (Tj) = S for every pair i;j of distinct integers with 1 � i;j� `. Chartrand et al. generalized the concept of connectivity as follows: The k-c潮nectivity , denoted byk(G), of G is defined byk(G) =minf�(S)g, where the minimum is taken over all k-subsets S of V (G). Thus �2(G) = �(G), where �(G) is the connectivity of G. Moreover, �n(G) is the maximum number of edge-disjoint spanning trees of G. This paper mainly focus on the k-connectivity of complete bipartite graphs Ka;b. First, we obtain the number of edge-disjoint spanning trees of Ka;b, which is b ab a+b−1 c, and specifically give the b ab a+b−1 c edge-disjoint spanning trees. Then based on this result, we get the k-connectivity of Ka;bfor all 2 � ka+b. Namely, if k > b−a+2
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.003 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".