Irregularity irr(G) = Σ |d(i)-d(j)| over edges. Albertson (1997). The simplest measure of how 'far from regular'. irr = 0 ⟺ graph is regular. Large irr = highly irregular. Maximum irregularity graphs are characterized.
How to Use
Select graph:
- irr: Irregularity
- |d-d|: Per edge
- =0?: Regular
Regularity Test
irr = 0 ⟺ regular. Instant test! irr is the L₁ version of degree disparity. Related measures: σ = Σ(dᵢ-dⱼ)² (L₂ version), GA gives L∞-style information.
Bounds
irr ≤ m·(Δ-δ). irr ≤ n²(Δ-δ)/4. For trees: irr maximized by star (irr = (n-2)(n-2)). Minimum non-zero: irr = 1 (one edge with |d-d|=1).
Step-by-Step Instructions
- 1Select graph.
- 2For each edge: |dᵢ-dⱼ|.
- 3Sum all terms.
- 4Check if irr=0.
- 5Quantify irregularity.