Modified first Zagreb ⁰M₁(G) = Σ 1/d(v)². Inverse-square of degree: low-degree vertices dominate. Nikolić et al. (2003). Complementary to M₁ = Σd². While M₁ emphasizes hubs, ⁰M₁ emphasizes peripheral atoms.
How to Use
Select graph:
- ⁰M₁: Modified Zagreb
- 1/d²: Per vertex
- vs M₁: Compare
⁰M₁ vs M₁
M₁ = Σd²: hub-centric. ⁰M₁ = Σ1/d²: leaf-centric. For star K_{1,n-1}: M₁ = (n-1)² + (n-1), ⁰M₁ = 1/(n-1)² + (n-1). Completely different perspectives on the same graph.
Bounds
⁰M₁(K_n) = n/(n-1)². ⁰M₁(star) ≈ n-1 (many leaves). ⁰M₁ ≥ n/Δ². For regular: ⁰M₁ = n/d².
Step-by-Step Instructions
- 1Select graph.
- 2For each vertex: 1/d(v)².
- 3Sum all terms.
- 4Compare with M₁.
- 5Assess leaf emphasis.