Discrete Isoperimetric Inequalities

One of the earliest problems in geometry is the isoperimetric problem, which was considered by the ancient Greeks. The problem is to find, among all closed curves of a given length, the one which encloses the maximum area. Isoperimetric problems for the discrete domain are in the same spirit but with different complexity. A basic model for communication and computational networks is a graph G = (V,E) consisting of a set V of vertices and a prescribed set E of unordered pairs of vertices. For a subset X of vertices, there are two types of boundaries: • The edge boundary ∂(X) = {{u, v} ∈ E : u ∈ X, v ∈ V \X}. • The vertex boundary δ(X) = {v ∈ V \X : {v, u} ∈ E for some u ∈ X}. Numerous questions arise in examining the relations between ∂(X), δ(X) and the sizes of X . Here the size of a subset of vertices may mean the number of vertices, the number of incident edges, or some other appropriate measure defined on graphs. In this paper, we will survey spectral techniques for studying discrete isoperimetric inequalities and the like. In addition, a number of applications in extremal graph theory and random walks will be included. This paper is organized as follows: