General sum-connectivity χₐ(G) = Σ (dᵢ+dⱼ)ᵃ. One parameter a unifies many indices: a=1 gives second Zagreb variant, a=-½ gives sum-connectivity χ, a=-1 gives harmonic/2. Zhou-Trinajstić (2009). The parametric approach reveals structural families.
How to Use
Select graph and exponent a:
- a=-½: Sum connectivity χ
- a=1: Zagreb variant
- Custom: Any real a
Special Cases
a=1: Σ(d+d) = M₁ (first Zagreb). a=-½: χ (sum-connectivity). a=-1: Σ1/(d+d) = H/2 (half harmonic). a=2: Σ(d+d)² = hyper-Zagreb. One formula, many indices!
Monotonicity in a
Increasing a: high-degree edges contribute more. Decreasing a: low-degree edges contribute more. At a=0: χ₀ = m (just edge count). Beautiful transition from hub-centric to leaf-centric.
Step-by-Step Instructions
- 1Select graph.
- 2Choose exponent a.
- 3For each edge: (dᵢ+dⱼ)ᵃ.
- 4Sum all terms.
- 5Compare different a values.