Recursive construction for 3-regular expanders

We present an algorithm which in n3(log n)3 time constructs a 3- regular expander graph on n vertices. In each step we substitute a pair of edges of the graph by a new pair of edges so that the total number of cycles of length s = [c log n] decreases (for some fixed absolute constant c). When we reach a local minimum in the number of cycles of length s the graph is an expander. The proof is completely elementary, we use only the basic results about the eigenvalues and eigenvectors of symmetric matrices.