On a conjecture of the harmonic index with given minimum degree of graphs and short proof of Liu’s result on Randić index
Bibliographic record
Abstract
The harmonic index H ( G ) of a graph G is defined as the sum of the weights 2 d u + d v of all edges u v of G , where d u denotes the degree of a vertex u . Delorme et al. [1] (2002) put forward a conjecture concerning the minimum Randić index among all connected graphs with n vertices and the minimum degree at least k . Motivated by this paper, a conjecture related to the minimum harmonic index among all connected graphs with n vertices and the minimum degree at least k was posed in [2]. In this work, we show that the conjecture is true for a connected graph on $n$ vertices with k vertices of degree n − 1 , and it is also true for a k -tree. Moreover, we give a shorter proof of Liu’s result [3].
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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 | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 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".