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. ∗Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Ramat Aviv, Tel Aviv, Israel. Research supported in part by a U.S.A.-Israeli BSF grant. †Department of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel