Geometric-arithmetic GA(G) = Σ 2√(dᵢdⱼ)/(dᵢ+dⱼ). The ratio GM/AM of endpoint degrees. By AM-GM: each term ≤ 1, so GA ≤ m. GA = m iff graph is regular. GA measures how 'balanced' each edge's endpoints are.
How to Use
Select graph:
- GA: GA index
- GM/AM: Per edge
- GA/m: Regularity
Regularity Measure
GA/m ∈ (0,1]. GA/m = 1 ⟺ regular graph. The closer to 1, the more 'balanced' the degree distribution across edges. GA/m measures local degree uniformity.
Bounds
GA ≤ m (equality iff regular). GA ≥ m·2√(δΔ)/(δ+Δ). For trees: star minimizes GA, path maximizes it. GA distinguishes regular from non-regular instantly.
Step-by-Step Instructions
- 1Select graph.
- 2For each edge: 2√(dᵢdⱼ)/(dᵢ+dⱼ).
- 3Sum all terms.
- 4Compute GA/m.
- 5Assess regularity.