Expander mixing lemma
The expander mixing lemma states that, for any two subsets of a d-regular expander graph with vertices, the number of edges between and is approximately what you would expect in a random d-regular graph, i.e. .
Let be a d-regular graph on n vertices with the second-largest eigenvalue (in absolute value) of the normalized adjacency matrix. 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
Let be the adjacency matrix for . For a vertex subset , let . Here is the standard basis element of with a one in the position. Thus in particular , and the number of edges between and is given by .
Expand each of and into a component in the direction of the largest-eigenvalue eigenvector and an orthogonal component:
where . Then
The conclusion follows, since , and .
Bilu and Linial 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 .
- Expander mixing lemma converse. 
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