Harmonic index H(G) = Σ 2/(d(i)+d(j)) over edges. Uses harmonic mean of endpoint degrees. Fajtlowicz (1987). H ≥ n/2·1/(δ+1). Correlates strongly with Randić index but sometimes outperforms it for QSAR predictions.
How to Use
Select graph:
- H: Harmonic index
- 2/(d+d): Per edge
- vs R: Compare Randić
Properties
Regular d-regular: H = m/d = nd/2d = n/2. Trees: H maximized by path (most uniform), minimized by star. H + Randić often bracket the true chemical property value.
Conjectures
Fajtlowicz conjecture: H ≥ r(G) (radius). Partially solved. Many open problems. H connects to independence number, chromatic number via degree-based arguments.
Step-by-Step Instructions
- 1Select graph.
- 2For each edge: 2/(dᵢ+dⱼ).
- 3Sum all terms.
- 4Compare with Randić.
- 5Apply to QSAR.