How to Use
Select graph:
- irr_t: Total irreg.
- vs A: Edge-only
- irr_t/A: Ratio
irr_t vs A
A = Σ_{edges} |d-d|. irr_t = Σ_{all pairs} |d-d|. irr_t ≥ A always. The difference irr_t - A measures irregularity contribution from NON-adjacent pairs. Global view vs local view.
Bounds
For n vertices with degree sequence d₁ ≥ d₂ ≥ ... ≥ dₙ: irr_t = Σ (2i-1-n)·dᵢ (rearrangement). Maximum for split-like graphs. irr_t ≤ n²(n-2)/4.
Step-by-Step Instructions
- 1Select graph.
- 2For ALL pairs (i,j): |dᵢ-dⱼ|.
- 3Sum all terms.
- 4Compare with A.
- 5Check irr_t/C(n,2).
Total Irregularity Calculator — Frequently Asked Questions
Why all pairs?+
Albertson only considers edge-adjacent mismatches. But non-adjacent vertices can also have very different degrees! irr_t captures this global information. Two hubs far apart contribute to irr_t but not to A.
irr_t from degree sequence?+
Yes! Sort degrees d₁ ≥ d₂ ≥ ... ≥ dₙ. Then irr_t = Σ_{i=1}^n (2i-1-n)·dᵢ. Only depends on sorted degrees! Graph structure beyond degrees doesn't affect irr_t.
irr_t always ≥ A?+
Yes! A sums over edges (a subset of all pairs). irr_t sums over all pairs (a superset). Every term in A appears in irr_t plus additional non-edge terms. irr_t ≥ A.
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