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 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 | 0.003 | 0.014 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.002 |
| Bibliometrics | 0.003 | 0.004 |
| Science and technology studies | 0.002 | 0.005 |
| Scholarly communication | 0.002 | 0.008 |
| Open science | 0.002 | 0.003 |
| Research integrity | 0.002 | 0.006 |
| Insufficient payload (model declined to judge) | 0.007 | 0.002 |
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".