The expander

Expander graph

In combinatorics, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion as described below...

 mixing lemma states that, for any two subsets  of a regular expander graph

Expander graph

In combinatorics, an expander graph is a sparse graph that has strong connectivity properties, quantified using vertex, edge or spectral expansion as described below...

 , the number of edges between and is approximately what you would expect in a random

Random graph

In mathematics, a random graph is a graph that is generated by some random process. The theory of random graphs lies at the intersection between graph theory and probability theory, and studies the properties of typical random graphs.-Random graph models:...

 d-regular graph

Regular graph

In graph theory, a regular graph is a graph where each vertex has the same number of neighbors; i.e. every vertex has the same degree or valency. A regular directed graph must also satisfy the stronger condition that the indegree and outdegree of each vertex are equal to each other...

, i.e. .

Statement

Let be a d-regular graph with (un-)normalized second-largest eigenvalue . Then for any two subsets , let denote the number of edges between S and T. We have


For a proof, see link in references.

Converse

Recently, Bilu and Linial

Nati Linial

Nathan Linial is an Israeli mathematician and computer scientist, a professor in the Rachel and Selim Benin School of Computer Science and Engineering at the Hebrew University of Jerusalem, and an ISI highly cited researcher....

showed that the converse holds as well: if a graph satisfies the conclusion of the expander mixing lemma, that is, for any two subsets ,


then its second-largest eigenvalue is .