The expander mixing lemma intuitively states that the edges of certain d-regular graphs are evenly distributed throughout the graph. In particular, the number of edges between two vertex subsets S and T is always close to the expected number of edges between them in a random d-regular graph, namely.

d-Regular Expander Graphs

Define an -graph to be a d-regular graph G on n vertices such that all of the eigenvalues of its adjacency matrix A_G except one have absolute value at most \lambda. The d-regularity of the graph guarantees that its largest absolute value of an eigenvalue is d. In fact, the all-1's vector \mathbf1 is an eigenvector of A_G with eigenvalue d, and the eigenvalues of the adjacency matrix will never exceed the maximum degree of G in absolute value. If we fix d and \lambda then -graphs form a family of expander graphs with a constant spectral gap.

Statement

Let G = (V, E) be an -graph. For any two subsets, let be the number of edges between S and T (counting edges contained in the intersection of S and T twice). Then

Tighter Bound

We can in fact show that using similar techniques.

Biregular Graphs

For biregular graphs, we have the following variation, where we take \lambda to be the second largest eigenvalue. Let be a bipartite graph such that every vertex in L is adjacent to d_L vertices of R and every vertex in R is adjacent to d_R vertices of L. Let with and. Let. Then Note that is the largest eigenvalue of G.

Proofs

Proof of First Statement

Let A_G be the adjacency matrix of G and let be the eigenvalues of A_G (these eigenvalues are real because A_G is symmetric). We know that \lambda_1=d with corresponding eigenvector, the normalization of the all-1's vector. Define and note that. Because A_G is symmetric, we can pick eigenvectors of A_G corresponding to eigenvalues so that forms an orthonormal basis of \mathbf R^n. Let J be the n\times n matrix of all 1's. Note that v_1 is an eigenvector of J with eigenvalue n and each other v_i, being perpendicular to, is an eigenvector of J with eigenvalue 0. For a vertex subset, let 1_U be the column vector with v^\text{th} coordinate equal to 1 if v\in U and 0 otherwise. Then, Let. Because A_G and J share eigenvectors, the eigenvalues of M are. By the Cauchy-Schwarz inequality, we have that. Furthermore, because M is self-adjoint, we can write This implies that and.

Proof Sketch of Tighter Bound

To show the tighter bound above, we instead consider the vectors and, which are both perpendicular to v_1. We can expand because the other two terms of the expansion are zero. The first term is equal to, so we find that We can bound the right hand side by using the same methods as in the earlier proof.

Applications

The expander mixing lemma can be used to upper bound the size of an independent set within a graph. In particular, the size of an independent set in an -graph is at most This is proved by letting T=S in the statement above and using the fact that e(S,S)=0. An additional consequence is that, if G is an -graph, then its chromatic number \chi(G) is at least d/\lambda. This is because, in a valid graph coloring, the set of vertices of a given color is an independent set. By the above fact, each independent set has size at most so at least d/\lambda such sets are needed to cover all of the vertices. A second application of the expander mixing lemma is to provide an upper bound on the maximum possible size of an independent set within a polarity graph. Given a finite projective plane \pi with a polarity \perp, the polarity graph is a graph where the vertices are the points a of \pi, and vertices x and y are connected if and only if In particular, if \pi has order q, then the expander mixing lemma can show that an independent set in the polarity graph can have size at most a bound proved by Hobart and Williford.

Converse

Bilu and Linial showed that a converse holds as well: if a d-regular graph G = (V, E) satisfies that for any two subsets with we have then its second-largest (in absolute value) eigenvalue is bounded by.

Generalization to hypergraphs

Friedman and Widgerson proved the following generalization of the mixing lemma to hypergraphs. Let H be a k-uniform hypergraph, i.e. a hypergraph in which every "edge" is a tuple of k vertices. For any choice of subsets of vertices,