Cut vertex and unicyclic graphs with the maximum number of connected induced subgraphs
Bibliographic record
Abstract
Cut vertices are often used as a measure of nodes’ importance within a network. These are nodes whose failure disconnects a connected graph. Let \(N(G)\) be the number of connected induced subgraphs of a graph \(G\). In this work, we investigate the maximum of \(N(G)\) where \(G\) is a unicyclic graph with \(n\) nodes of which \(c\) are cut vertices. For all valid \(n,c\), we give a full description of those maximal (that maximise \(N(.)\)) unicyclic graphs. It is found that there are generally two maximal unicyclic graphs. For infinitely many values of \(n,c\), however, there is a unique maximal unicyclic graph with \(n\) nodes and \(c\) cut vertices. In particular, the well-known negative correlation between the number of connected induced subgraphs of trees and the Wiener index (sum of distances) fails for unicyclic graphs with \(n\) nodes and \(c\) cut vertices: for instance, the maximal unicyclic graph with \(n=3,4\mod 5\) nodes and \(c=n-5>3\) cut vertices is different from the unique graph that was shown by Tan et al. [The Wiener index of unicyclic graphs given number of pendant vertices or cut vertices. J. Appl. Math. Comput., 55:1–24, 2017] to minimise the Wiener index. Our main characterisation of maximal unicyclic graphs with respect to the number of connected induced subgraphs also applies to unicyclic graphs with \(n\) nodes, \(c\) cut vertices and girth at most \(g>3\), since it is shown that the girth of every maximal graph with \(n\) nodes and \(c\) cut vertices cannot exceed \(4\).
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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 | 0.007 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.002 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.002 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| 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".