Counting Spanning Out-trees in Multidigraphs
Bibliographic record
Abstract
This paper generalizes an inclusion/exclusion counting formula of Temperley for the number of spanning trees of a graph based on its complement. The new formula is for the number of out-trees of a digraph which may have multiple arcs. This provides an extension of Temperley's formula to graphs with multiple edges. Determining which graphs have a maximum number of spanning trees is important for network reliability applications. Some examples are given to show how considering the underlying digraphs of undirected graphs using the new formula simplifies comparisons as terms which correspond to arcs in the digraphs can cancel. Department of Computer Science, University of Victoria, Victoria, B.C. V8W 3P6, CANADA, wendym@csr.uvic.ca. Research supported by NSERC grant OGP0041927 y Department of Mathematics and Statistics, Simon Fraser University, 8888 University Dr., Burnaby, B. C., V5A 1S6, CANADA, klwood@sfu.ca 1 2 1 Introduction An undirected graph G consists of a set V of...
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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 | 0.001 | 0.008 |
| 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 | 0.002 | 0.005 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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".