Random Cayley graphs and expanders

For every 1 > δ > 0 there exists a c = c (δ) > 0 such that for every group G of order n , and for a set S of c (δ) log n random elements in the group, the expected value of the second largest eigenvalue of the normalized adjacency matrix of the Cayley graph X(G, S) is at most (1 ‐ δ). This implies that almost every such a graph is an ϵ(δ)‐expander. For Abelian groups this is essentially tight, and explicit constructions can be given in some cases.