RANDOM CAYLEY-GRAPHS AND EXPANDERS

For every 1 > delta > 0 there exists a c = c(delta) > 0 such that for every group G of order n, and for a set S of c(delta) 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 - delta). This implies that almost every such a graph is an epsilon(delta)-expander. For Abelian groups this is essentially tight, and explicit constructions can be given in some cases. (C) 1994 John Wiley & Sons, Inc.