General harmonic Hₐ(G) = Σ (2/(dᵢ+dⱼ))ᵃ. Zhong (2012). a=1: classical harmonic H. a=2: squared harmonic. a=-1: Σ(d+d)/2 = M₁/2. Parametric family bridging harmonic and Zagreb perspectives.
How to Use
Select graph and exponent a:
- a=1: Harmonic H
- a=-1: M₁/2
- Custom: Any real a
Special Cases
a=1: H (harmonic). a=2: H² (squared harmonic). a=-1: M₁/2. a=0: m (edge count). As a→∞: dominated by minimum degree-sum edge. Continuous interpolation.
Behavior in a
For a>0: low degree-sum edges dominate (their 2/(d+d) term is largest). For a<0: high degree-sum edges dominate. a=0: all edges equal. The parameter a controls the 'focus' on edge types.
Step-by-Step Instructions
- 1Select graph.
- 2Choose exponent a.
- 3For each edge: (2/(dᵢ+dⱼ))ᵃ.
- 4Sum all terms.
- 5Explore behavior in a.