These are partly due to a construction of an expander graph.

The following are some properties of expander graphs that have proven useful in many areas.

This definition is used in construction of expander graphs.

It can be seen as a special case of Cheeger inequalities in expander graphs.

It has been shown that nontrivial lossless expander graphs exist.

A key requirement when designing such protocols is that the neighbor set trace out an expander graph.

Zemor's algorithm is based on a type of expander graphs called Tanner graph.

Let be an expander graph with normalized second-largest eigenvalue .

Then is a expander graph if every small enough subset , has the property that has at least distinct neighbors in .

Expander graph (discrete case)

Introduction to Expander Graphs.

The Cheeger constant is especially important in the context of expander graphs as it is a way to measure the edge expansion of a graph.

Margulis gave the first construction of expander graphs, which was later generalized in the theory of Ramanujan graphs.

In coding theory, expander codes form a class of error-correcting codes that are constructed from bipartite expander graphs.

The expander walk sampling lemma, due to and , states that this also holds true when sampling from a walk on an 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.

In 2005 Irit Dinur discovered a different proof of the PCP theorem, using expander graphs.

In graph theory, isoperimetric inequalities are at the heart of the study of expander graphs, which are sparse graphs that have strong connectivity properties.

The first of these sections provides an alphabetized set of articles on 99 specific mathematical concepts such as the axiom of choice, expander graphs, and Hilbert space.

In coding theory, an expander code is a linear block code whose parity check matrix is the adjacency matrix of a bipartite expander graph.

These codes have good relative distance , where and are properties of the expander graph as defined later), rate , and decodability (algorithms of running time exist).

In 2002 Omer Reingold, Salil Vadhan, and Avi Wigderson have given a simple, explicit combinatorial construction of constant-degree expander graphs.

His recent work has included approximate counting and volume computation via random walks; finding edge disjoint paths in expander graphs, and exploring anti-Ramsey theory and the stability of routing algorithms.

In 2008 Linial and his co-authors won the Levi L. Conant Prize of the American Mathematical Society for best mathematical exposition for this article, a survey on expander graphs.

The expander mixing lemma states that, for any two subsets of a d-regular expander graph , the number of edges between and is approximately what you would expect in a random d-regular graph, i.e. .