Hyper Zagreb Index Calculator

squared degree-sum edges

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About Hyper Zagreb Index Calculator

A hyper-Zagreb index calculator computing HM(G) = Σ_{(i,j)∈E} (d(i)+d(j))². Shirdel-Rezapour-Sayadi (2013). Squares the degree sum per edge. HM = M₁ + 2·M₂ for simple graphs. More discriminating than M₁ or M₂ alone. Client-side.

Hyper Zagreb Index Calculator Features

  • HM(G)
  • (d+d)²
  • M₁+2M₂
  • Shirdel '13
  • Common graphs
Hyper-Zagreb HM(G) = Σ (dᵢ+dⱼ)² over edges. Squares the degree sum: amplifies high-degree edge contributions. HM = M₁ + 2M₂ relates to both Zagreb indices. Shirdel-Rezapour-Sayadi (2013). More discriminating than either Zagreb index alone.

How to Use

Select graph:

  • HM: Hyper-Zagreb
  • (d+d)²: Per edge
  • M₁+2M₂: Verify

Zagreb Relation

HM = Σ(dᵢ+dⱼ)² = Σ(dᵢ²+2dᵢdⱼ+dⱼ²) = M₁ + 2M₂. Beautiful decomposition: hyper-Zagreb combines both Zagreb indices into one! Captures both vertex and edge branching.

Bounds

HM ≥ 8m³/n² (sharp for regular). HM(K_n) = n(n-1)(2n-2)² / 2. For trees: HM maximized by path, minimized by star.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2For each edge: (dᵢ+dⱼ)².
  3. 3Sum all terms.
  4. 4Verify HM=M₁+2M₂.
  5. 5Compare graphs.

Hyper Zagreb Index Calculator — Frequently Asked Questions

Why square the degree sum?+

Squaring amplifies large values: edge between degree-10 vertices contributes 400, edge between degree-2 vertices contributes 16. This 25× amplification makes HM very sensitive to structural differences.

How does HM relate to Zagreb indices?+

HM = M₁ + 2M₂ exactly. So if you know M₁ and M₂, you get HM for free. But HM as a single number is more discriminating than either M₁ or M₂ individually.

Is HM better than M₁ and M₂?+

For discrimination power: yes. HM distinguishes more non-isomorphic graphs. For QSAR: depends on the property. HM sometimes outperforms both for predicting boiling points.

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