Definition
Vertex expansion: |N(S)| ≥ c |S| for all small S. Edge expansion: |E(S, V\S)| ≥ h |S|.
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Cheeger's inequality
Spectral gap λ₂ ≈ edge expansion h. Formal: λ₂/2 ≤ h ≤ √(2 λ₂).
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Constructions
Random d-regular: expander w.h.p. Ramanujan graphs: optimal expansion (LPS construction).
Applications
Error-correcting codes. Pseudorandomness. Hash function analysis. Distributed algorithms.