Expanders from Epsilon-Biased Sets

Expander graphs informally are graphs that given any subset S that is not too large, the set of vertices connected to S contains a large number of vertices outside of S. There are many constructions and applications for expander graphs leading to entire courses on the subject.

The adjacency matrix A of a graph G of n vertices is an n×n matrix such that a_i,j is 1 if there is an edge between vertices i and j and 0 otherwise. Noga Alon noticed that a graph that has a large gap between the first and second eigenvalue of the adjacency matrix will be a good expander.

We can use ε-biased sets to get expanders. Let S be a ε-biased set for F^m for F the field of 2 elements. Consider the graph G consisting of 2^m vertices labelled with the elements of F^m and an edge from x to y if y=x+s or x=y+s. This kind of graph G is known as a Cayley graph.

By looking at the eigenvalues the adjacency matrix A of G we can show G is an expander. The eigenvectors v are just the vectors corresponding to the functions g in L described earlier. For any vector a we have

(Ag)(a) = Σ_{s in S} g(a+s) = g(a) Σ_{s in S} g(s) since g(a+s) = g(a)g(s). Let g(S) = Σ_{s in S} g(s). We now have that Ag = g(S) g. So g is an eigenvector with eigenvalue g(S). If g is the constant one function then g(S)=|S|. Since S is an ε-biased set, g(S)≤ε|S| for every other g, so the second eigenvalue is much smaller than the largest eigenvalue and G must be an expander.