The adjacency matrix A of a graph G of n vertices is an n×n matrix such that ai,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 Fm for F the field of 2 elements. Consider the graph G consisting of 2m vertices labelled with the elements of Fm 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