A study of Fibonacci cordial labeling in structured graph families
Bibliographic record
Abstract
<p>A <em>Fibonacci cordial labeling</em> of a graph <span class="math inline">\(G\)</span> is an injective function <span class="math inline">\(f: V(G) \rightarrow \{F_0, F_1, \dots,\\ F_n\}\)</span>, where <span class="math inline">\(F_i\)</span> denotes the <span class="math inline">\(i^{\text{th}}\)</span> Fibonacci number, such that the induced edge labeling <span class="math inline">\(f^*: E(G) \rightarrow \{0,1\}\)</span>, given by <span class="math inline">\(f^*(uv) = (f(u) + f(v))\)</span> <span class="math inline">\((\bmod\ 2)\)</span>, satisfies the balance condition <span class="math inline">\(|e_f(0) - e_f(1)| \le 1\)</span>. Here, <span class="math inline">\(e_f(0)\)</span> and <span class="math inline">\(e_f(1)\)</span> represent the number of edges labeled 0 and 1, respectively. A graph that admits such a labeling is termed a <em>Fibonacci cordial graph</em>. In this paper, we investigate the existence and construction of Fibonacci cordial labelings for several families of graphs, including <em>Generalized Petersen graphs</em>, <em>open and closed helm graphs</em>, <em>joint sum graphs</em>, and <em>circulant graphs of small order</em>. New results and examples are presented, contributing to the growing body of knowledge on graph labelings inspired by numerical sequences.</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 | 0.002 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| 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".