Cheeger constant h(G): minimum edge cut normalized by volume. h = min |E(S,S̄)| / min(vol(S),vol(S̄)). The Cheeger inequality λ₂/2 ≤ h ≤ √(2λ₂) is one of the most powerful tools in spectral graph theory, connecting eigenvalues to graph structure.
How to Use
Select graph:
- h: Cheeger const.
- λ₂: Algebraic conn.
- Ineq: λ₂/2≤h≤√(2λ₂)
Cheeger Inequality
λ₂/2 ≤ h(G) ≤ √(2λ₂). Left: spectral gap lower-bounds expansion. Right: expansion upper-bounds spectral gap. Computable bridge between two fundamental properties.
Applications
Image segmentation (normalized cuts = Cheeger minimization). Community detection. MCMC convergence bounds. Network bottleneck identification. Spectral clustering.
Step-by-Step Instructions
- 1Select graph.
- 2Compute h.
- 3Verify Cheeger inequality.
- 4Compare with λ₂.
- 5Find bottleneck.