Collatz-Sinogowitz CS(G) = λ₁ - 2m/n. The first irregularity index ever proposed (1957)! λ₁ = spectral radius (largest eigenvalue of adjacency matrix). CS ≥ 0 always. CS = 0 ⟺ regular. Spectral approach to irregularity.
How to Use
Select graph:
- CS: Spectral irreg.
- λ₁: Spectral radius
- d̄: Average degree
Spectral View
For regular d-graphs: λ₁ = d = d̄, so CS = 0. For irregular: λ₁ > d̄ always (λ₁ ≥ d̄ by Rayleigh quotient). The excess λ₁-d̄ measures how much the dominant eigenstructure deviates from regularity.
Historical
Proposed by Collatz & Sinogowitz in 1957 — the dawn of spectral graph theory! Predates all other irregularity measures by decades. Elegant: uses one number (spectral radius) to measure graph structure.
Step-by-Step Instructions
- 1Select graph.
- 2Compute λ₁ (spectral radius).
- 3Compute d̄ = 2m/n.
- 4CS = λ₁ - d̄.
- 5Check if CS≈0.