Modified first Zagreb mM₁(G) = Σ 1/d(v)². Nikolić et al. (2003). Inverts the classical M₁ = Σd²: instead of penalizing high degree, mM₁ rewards low degree. Pendant vertices contribute 1, hub vertices contribute ~0. Leaf-centric index.
How to Use
Select graph:
- mM₁: Modified Zagreb
- 1/d²: Per vertex
- vs M₁: Compare
Inversion of M₁
M₁ = Σd². mM₁ = Σ1/d². Product M₁·mM₁ ≥ n² (Cauchy-Schwarz). For regular: M₁·mM₁ = n² exactly. The inequality measures how far from regular.
Bounds
n/Δ² ≤ mM₁ ≤ n/δ². For K_n: mM₁ = n/(n-1)² → 0 as n→∞. For star: mM₁ = 1/(n-1)² + (n-1) ≈ n-1 (leaves dominate).
Step-by-Step Instructions
- 1Select graph.
- 2For each vertex: 1/d(v)².
- 3Sum all terms.
- 4Compare with M₁.
- 5Check M₁·mM₁ ≥ n².