Symmetric division degree SDD(G) = Σ (dᵢ/dⱼ + dⱼ/dᵢ) over edges. By AM-GM: each term ≥ 2, so SDD ≥ 2m. SDD = 2m iff regular. Measures how 'balanced' degree ratios are across edges. Vukičević-Gašperov (2010).
How to Use
Select graph:
- SDD: Sym. div. deg.
- d/d+d/d: Per edge
- SDD/2m: Regularity
Regularity Measure
SDD/2m = 1 ⟺ regular. SDD/2m > 1: degree imbalance. The ratio quantifies departure from regularity. Each edge contributes exactly 2 if endpoints have equal degree.
Bounds
SDD ≥ 2m (regular). SDD ≤ m·(Δ/δ + δ/Δ). Tight for regular and bi-regular graphs. SDD relates to M₁: SDD = Σ(d²+d²)/(dd) = F*/M₂ variant.
Step-by-Step Instructions
- 1Select graph.
- 2For each edge: dᵢ/dⱼ + dⱼ/dᵢ.
- 3Sum all terms.
- 4Compute SDD/2m.
- 5Assess regularity.