Abstract:
For every 1 ? ffi ? 0 there exists a c = c(ffi) ? 0 such that for every group G of order n, and for a set S of c(ffi) 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\Gammaffi). This implies that almost every such a graph is an "(ffi)-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. y Department of Mathematics, Hebrew University of Jerusalem, Givat Ram, Jerusalem, Israel 0 1.
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