Cut vertex and unicyclic graphs with the maximum number of connected induced subgraphs
Bibliographic record
Abstract
<p>Cut vertices are often used as a measure of nodes’ importance within a network. These are nodes whose failure disconnects a connected graph. Let <span class="math inline">\(N(G)\)</span> be the number of connected induced subgraphs of a graph <span class="math inline">\(G\)</span>. In this work, we investigate the maximum of <span class="math inline">\(N(G)\)</span> where <span class="math inline">\(G\)</span> is a unicyclic graph with <span class="math inline">\(n\)</span> nodes of which <span class="math inline">\(c\)</span> are cut vertices. For all valid <span class="math inline">\(n,c\)</span>, we give a full description of those maximal (that maximise <span class="math inline">\(N(.)\)</span>) unicyclic graphs. It is found that there are generally two maximal unicyclic graphs. For infinitely many values of <span class="math inline">\(n,c\)</span>, however, there is a unique maximal unicyclic graph with <span class="math inline">\(n\)</span> nodes and <span class="math inline">\(c\)</span> 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 <span class="math inline">\(n\)</span> nodes and <span class="math inline">\(c\)</span> cut vertices: for instance, the maximal unicyclic graph with <span class="math inline">\(n=3,4\mod 5\)</span> nodes and <span class="math inline">\(c=n-5>3\)</span> cut vertices is different from the unique graph that was shown by Tan et al. [<span><em>The Wiener index of unicyclic graphs given number of pendant vertices or cut vertices</em></span>. 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 <span class="math inline">\(n\)</span> nodes, <span class="math inline">\(c\)</span> cut vertices and girth at most <span class="math inline">\(g>3\)</span>, since it is shown that the girth of every maximal graph with <span class="math inline">\(n\)</span> nodes and <span class="math inline">\(c\)</span> cut vertices cannot exceed <span class="math inline">\(4\)</span>.</p>
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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 | 0.003 | 0.001 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.000 |
| Bibliometrics | 0.000 | 0.002 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".