Lecture 17: Expander Graphs 1 Overview of Expander Graphs

Let G = (V,E) be an undirected d-regular graph, here, |V | = n, deg(u) = d for all u ∈ V . We will typically interpret the properties of expander graphs in an asymptotic sense. That is, there will be an infinite family of graphs G, with a growing number of vertices n. By “sparse”, we mean that the degree d of G should be very slowly growing as a function of n. When n goes to infinity (n → ∞), d is thought as a constant, so the graph automatically becomes sparse as n grows large, since the number of edges |E| = d 2 n ∼ O(n). The “highlyconnected” property has a variety of different interpretations, like in terms of edge expansion, vertex expansion or spectral expansion. Informally, we define a graph is well-connectness that for every S ⊆ V , S is a not-too-large set of vertices (say 0 < |S| ≤ n 2 ) but has lots of (say Ω(|S|)) edges for edge expansion or vertices for vertex expansion on its boundary.