Notes 3 : Expander graphs – a ubiquitous pseudorandom structure

In this lecture, we will focus on expander graphs (also called expanders), which are pseudoran-dom objects in a more restricted sense than what we saw in the last two lectures. The reader is also referred to the monograph [1] and the tutorial slides [2] for more detailed surveys of today's topics. Expander graphs are universally useful in computer science and have many applications in de-randomization, circuit complexity, error correcting codes, communication and sorting networks, approximate counting, computational information, data structures, etc. Expander graphs also have many interesting applications in various areas of pure mathematics: topology, group theory, measure theory, number theory and especially graph theory. 1 Expander graphs: definition and basic properties Expander graphs are graphs with an additional special " expansion " property. This property can be equivalently described combinatorically, geometrically, probabilistically, or algebraically. 1. Combinatoric formulation: Expander graphs are sparse d-regular graph (for some fixed d), but highly connected; any two disjoint sets of vertices cannot be disconnected from each other without removing a lot of edges (i.e., no small cuts). In other words, a d-regular graph G with n vertices is expander if for any subset S of vertices satisfying |S| < n/2, number of edges connecting S and the complement of S is at least α|S|d for a constant α. 2. Geometric formulation: Expander graphs have high isoperimetry, i.e., any proper subset S of vertices has a large boundary ∂S, where ∂S is defined as the set of vertices which are not in S but are adjacent to a vertex in S. 3. Probabilistic formulation: A random walk on an expander graph converges very rapidly (in O(log n) steps) to the uniform distribution. 4. Algebraic formulation: If we look at the the adjacency matrix of an expander graph, then there is a large gap between second largest eigenvalue and the largest eigenvalue, which in this case equals the degree of the graph. Since all of these formulations of the expansion property are equivalent, we will use the algebraic formulation in this lecture. Let G = (V, E) be a d-regular graph with n vertices. We let A G denote the normalized adjacency matrix of G, i.e., It easy to check that if we sort the eigenvalues λ i of A G in non-increasing order, then 1 ≥ λ i ≥ −1 and the largest eigenvalue λ 1 = 1. We will let λ(G) denote the second …