Signless Laplacian spectral determinations of the starlike trees <span class="math inline">\(ST(n, d_1)\)</span> and the double starlike trees <span class="math inline">\(H_n(p, p)\)</span>
Bibliographic record
Abstract
<p>Two graphs are said to be <span class="math inline">\(Q\)</span>-cospectral (respectively, <span class="math inline">\(A\)</span>-cospectral) if they have the same signless Laplacian (respectively, adjacency) spectrum. A graph is said to be <span class="math inline">\(DQS\)</span> (respectively, <span class="math inline">\(DAS\)</span>) if there is no other non-isomorphic graphs <span class="math inline">\(Q\)</span>-cospectral (respectively, <span class="math inline">\(A\)</span>-cospectral) with it. A tree on <span class="math inline">\(n\)</span> vertices with maximum degree <span class="math inline">\(d_1\)</span> is called starlike and denoted by <span class="math inline">\(ST(n, d_1)\)</span>, if it has exactly one vertex with the degree greater than 2. A tree is called double starlike if it has exactly two vertices of degree greater than 2. If we attach <span class="math inline">\(p\)</span> pendant vertices (vertices of degree 1) to each of pendant vertices of a path <span class="math inline">\(P_n\)</span>, the the resulting graph is called the double starlike tree <span class="math inline">\(H_n(p,p)\)</span>. In this article, we prove that all double starlike trees <span class="math inline">\(H_n(p,p)\)</span> are <span class="math inline">\(DQS\)</span>, where <span class="math inline">\(p\geq 1, n\geq 2\)</span> and <span class="math inline">\(p\)</span> denotes . In addition, by a simple method, we show that all starlike trees are <span class="math inline">\(DQS\)</span> excluding <span class="math inline">\(K_{1,3}=ST(4,3)\)</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 | 0.004 | 0.001 |
| Meta-epidemiology (narrow) | 0.003 | 0.002 |
| Meta-epidemiology (broad) | 0.004 | 0.002 |
| Bibliometrics | 0.002 | 0.007 |
| Science and technology studies | 0.004 | 0.005 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.005 | 0.002 |
| Research integrity | 0.002 | 0.003 |
| 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; both teacher heads agree on what is shown here.
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".