From the Spectrum of the Adjacency Matrix to the Spectrum of Directed Edge Matrix: Counting Cycles of a Bipartite Graph Through a Simple Equation
Bibliographic record
Abstract
Counting short cycles in bipartite graphs is a fundamental problem of interest in many fields including the analysis and design of low-density parity-check (LDPC) codes. There are two computational approaches to count short cycles (with length smaller than 2g, where g is the girth of the graph) in bipartite graphs. The first approach is applicable to a general (irregular) bipartite graph, and uses the spectrum {ηi} of the directed edge matrix of the graph to compute the multiplicity Nk of k-cycles with kk=Σiηik/(2k). This approach has a computational complexity O(|E|3), where |E| is number of edges in the graph. The second approach is only applicable to bi-regular bipartite graphs, and uses the spectrum {λi} of the adjacency matrix (graph spectrum) and the degree sequences of the graph to compute Nk. The complexity of this approach is O(|V|3), where |V| is number of nodes in the graph. This complexity is less than that of the first approach, but the equations involved in the computations of the second approach are very tedious, particularly for k ≥ g + 6. In this paper, we establish an analytical relationship between the two spectra {ηi} and {λi} for bi-regular bipartite graphs. Through this relationship, the former spectrum can be derived from the latter through simple equations. This allows the computation of Nkusing Nk= Σiηik/(2k) but with a complexity of O(|V|3) rather than O(|E|3).
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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 | 0.009 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.003 | 0.002 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.004 |
| Open science | 0.002 | 0.001 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.006 | 0.001 |
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".