Second Hyper Zagreb Calculator

squared degree-product sum

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About Second Hyper Zagreb Calculator

A second hyper-Zagreb calculator computing HM₂(G) = Σ (dᵢ·dⱼ)² over edges. Shirdel-Rezapour-Sayadi (2013). Squared products: amplifies hub-hub connections. HM₂ vs HM₁: product² vs sum². For regular d-graphs: HM₂ = m·d⁴. Client-side.

Second Hyper Zagreb Calculator Features

  • HM₂(G)
  • Σ(dd)²
  • Quartic
  • Shirdel '13
  • Common graphs
Second hyper-Zagreb HM₂(G) = Σ (dᵢ·dⱼ)² over edges. Complement to HM₁ = Σ(dᵢ+dⱼ)². Shirdel-Rezapour-Sayadi (2013). Quartic in degree: d⁴ growth for regular graphs. Extreme sensitivity to high-degree endpoints.

How to Use

Select graph:

  • HM₂: Hyper-Zagreb 2nd
  • (dd)²: Per edge
  • vs HM₁: Compare

HM₂ vs HM₁

HM₁ = Σ(d+d)². HM₂ = Σ(dd)². By AM-GM: (d+d)² ≥ 4dd, but (dd)² vs (d+d)² depends on d values. For regular: HM₂/HM₁ = d⁴/(4d²) = d²/4.

Growth Rate

HM₂ grows as d⁴ for regular graphs. Fastest-growing of the standard Zagreb family. Extremely hub-sensitive: a single hub-hub edge can dominate the entire sum.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2For each edge: (dᵢ·dⱼ)².
  3. 3Sum all terms.
  4. 4Compare with HM₁.
  5. 5Identify dominant edges.

Second Hyper Zagreb Calculator — Frequently Asked Questions

HM₂ vs M₂?+

M₂ = Σdd (linear product). HM₂ = Σ(dd)² (squared product). HM₂ amplifies: edge between degree-10 vertices contributes 100 to M₂ but 10000 to HM₂. Quartic sensitivity.

When HM₂ useful?+

When hub-hub connections are critical: protein-protein interaction networks, social network core detection, power grid vulnerability (high-degree hub connections are most critical).

For trees?+

Trees: only pendant edges (d=1) and internal edges. HM₂(star) = (n-1)·(1·(n-1))² = (n-1)³. HM₂(path) = 2·(1·2)² + (n-3)·(2·2)² = 8 + 16(n-3).

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