Sum Connectivity Index Calculator

degree-sum reciprocal root

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About Sum Connectivity Index Calculator

A sum connectivity index calculator computing χ(G) = Σ_{(i,j)∈E} 1/√(d(i)+d(j)). Zhou-Trinajstić (2009). Uses degree sum instead of product (cf. Randić). χ and R often give complementary QSAR predictions. Better for some molecular properties. Client-side.

Sum Connectivity Index Calculator Features

  • χ(G)
  • 1/√(d+d)
  • vs Randić
  • Zhou '09
  • Common graphs
Sum connectivity χ(G) = Σ 1/√(dᵢ+dⱼ) over edges. Zhou-Trinajstić (2009). Uses degree sum instead of Randić's product. By AM-GM: dᵢ+dⱼ ≥ 2√(dᵢdⱼ), so χ ≤ R. Complementary to Randić for QSAR applications.

How to Use

Select graph:

  • χ: Sum conn.
  • 1/√(d+d): Per edge
  • χ≤R: Compare

χ vs Randić

Randić: 1/√(dᵢ·dⱼ). Sum: 1/√(dᵢ+dⱼ). By AM-GM: dᵢ+dⱼ ≥ 2√(dᵢdⱼ) → 1/√(dᵢ+dⱼ) ≤ 1/√(2√(dᵢdⱼ)). Sum is always ≤ Randić. Different perspective on edge connectivity.

Bounds

χ ≤ R always. For regular d-regular: χ = m/√(2d) = n√(d)/(2√2). χ(star) = (n-1)/√n. χ(path) computed edge-by-edge.

Step-by-Step Instructions

  1. 1Select graph.
  2. 2For each edge: 1/√(dᵢ+dⱼ).
  3. 3Sum all terms.
  4. 4Verify χ ≤ R.
  5. 5Compare QSAR.

Sum Connectivity Index Calculator — Frequently Asked Questions

Why sum instead of product?+

Product √(dᵢdⱼ) is the geometric mean of degrees. Sum (dᵢ+dⱼ) is proportional to arithmetic mean. Different means capture different structural aspects. Sum is more appropriate when degree contributions are additive.

Is χ always less than R?+

Yes! By AM-GM inequality: dᵢ+dⱼ ≥ 2√(dᵢdⱼ), so 1/√(dᵢ+dⱼ) ≤ 1/√(2√(dᵢdⱼ)) ≤ 1/√(dᵢdⱼ). Therefore χ ≤ R with equality iff all edges have dᵢ=dⱼ (regular graph).

When does χ outperform R?+

χ sometimes better predicts properties that depend on the total connectivity environment (sum of neighbor degrees) rather than multiplicative interactions. Tested empirically for various molecular families.

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