Lecture 8: Spectral Expansion Ii

Remark Here is an interpretation of Theorem 2. Consider the following two random experiments. Experiment 1: pick a random vertex u ∈ V of the graph G, and then pick one of its d neighbors v, uniformly at random. Experiment 2: pick a random vertex u ∈ V and then pick a random vertex v ∈ V . What is the probability of picking an ordered pair (u, v) such that u ∈ B and v ∈ C? For Experiment 1, it is e(B,C) nd ; for experiment 2, it is μ(B)μ(C). So, Theorem 2 says that, for an expander graph G, Experiment 1 results in a pair from B×C with a probability that is a constant fraction (depending on λ) of the probability of picking such a pair in Experiment 2. For expander graphs where we have a bound on λ2(G) rather than just λ2, we can prove that the probability of getting a pair from B × C in Experiment 1 is close to that in Experiment 2 — see the Expander Mixing Lemma below (Lemma 5).