Sigma Index Calculator

squared degree deviation

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About Sigma Index Calculator

A sigma index calculator computing σ(G) = Σ_{(i,j)∈E} (d(i)-d(j))². Gutman-Togan (2021). L₂ irregularity measure. σ = 0 iff regular. σ = M₁+M₂-2M₂ = F - 2M₂ + M₁ (via Zagreb). More sensitive to degree outliers than Albertson irr. Client-side.

Sigma Index Calculator Features

  • σ(G)
  • (dᵢ-dⱼ)²
  • L₂ irr.
  • 0 iff reg.
  • Common graphs
Sigma index σ(G) = Σ (d(i)-d(j))² over edges. L₂ (squared) version of irregularity. σ = 0 ⟺ regular. Quadratic penalty: large degree differences contribute disproportionately. σ = F - 2M₂ connects to forgotten and Zagreb indices.

How to Use

Select graph:

  • σ: Sigma index
  • (d-d)²: Per edge
  • =0?: Regular

L₂ vs L₁

Albertson irr: Σ|d-d| (L₁). Sigma σ: Σ(d-d)² (L₂). L₂ penalizes outliers more. Edge with |Δd|=10: irr contributes 10, σ contributes 100. σ is the 'variance' of degree along edges.

Zagreb Connection

σ = Σ(dᵢ-dⱼ)² = Σ(dᵢ²+dⱼ²) - 2Σdᵢdⱼ = F - 2M₂. So σ is determined by F (forgotten) and M₂ (second Zagreb). Three indices are interrelated!

Step-by-Step Instructions

  1. 1Select graph.
  2. 2For each edge: (dᵢ-dⱼ)².
  3. 3Sum all terms.
  4. 4Check σ=F-2M₂.
  5. 5Compare with irr.

Sigma Index Calculator — Frequently Asked Questions

Why squared differences?+

Squaring penalizes large deviations: edge with |Δd|=5 contributes 25 to σ but only 5 to irr. σ captures 'variance' of degree along edges. More informative for detecting structural outliers.

σ = F - 2M₂: what does this mean?+

Forgotten index F = Σ(dᵢ²+dⱼ²) over edges. Second Zagreb M₂ = Σdᵢdⱼ. Their difference captures squared deviation. So σ is redundant if you know F and M₂!

When σ >> irr²/m?+

σ >> irr²/m means the degree differences are concentrated on few edges (not spread equally). Small σ/irr² ratio: uniform irregularity. Large ratio: localized irregularity spikes.

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