Isoperimetric number i(G): minimum boundary-to-volume ratio over all small subsets. i(G) = min_{|S|≤n/2} |∂S|/|S|. High i → good expander graph. Low i → bottleneck exists. Fundamental for random walk mixing, error-correcting codes, network design.
How to Use
Select graph:
- i: Isoperimetric #
- |∂S|/|S|: Ratio
- Expand: Quality
Expander Graphs
Family with i(G) ≥ c > 0 as n→∞. Optimal for networks: every small set has many boundary neighbors. Used in: derandomization, coding theory, compressed sensing, cryptocurrency.
Cheeger Inequality
λ₂/2 ≤ i(G) ≤ √(2λ₂) where λ₂ = algebraic connectivity. Bridges spectral and combinatorial expansion. One of the most important results in spectral graph theory.
Step-by-Step Instructions
- 1Select graph.
- 2Find min |∂S|/|S|.
- 3Evaluate expansion.
- 4Apply Cheeger.
- 5Compare graphs.