2005 ) Lecture 3 : Expander Graphs and PCP Theorem Proof Overview

Recall in last lecture that we defined a (n, d, λ)-expander to be a d-regular n-vertex undirected graph with second eigenvalue λ. We also defined the edge expansion of a graph G with vertex set V to be φ(G) = min S⊂V |S|≤n/2 |E(S, S)| |S| , where E(S, S) is the set of edges between a vertex set S and its complement. The following lemma shows that the eigenvalue formulation of an expander is essentially equivalent to edge expansion. Lemma 1.1. Let G be a (n, d, λ) expander. Then φ(G) ≥ (d − λ)/2. Remark 1.2. In the other, harder, direction, it is possible to show that φ(G) ≤ 2d(d − λ). For our purposes of constructing and using expanders, the easier direction shown in this lemma is enough. Proof. Let V and E be the vertex and edge sets of G. Let S ⊂ V with |S| ≤ n/2. We will set up a vector x that is similar to the characteristic vector of S, but perpendicular to − → 1. Then by the Rayleigh coefficient formulation of λ, we have that Ax ≤ λx, where A = A(G) is the adjacency matrix of G.