 
Summary: Random Cayley Graphs and Expanders
Noga Alon
Yuval Roichman
February 22, 2002
Abstract
For every 1 > > 0 there exists a c = c() > 0 such that for every
group G of order n, and for a set S of c() 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). This implies that almost every such a graph is an ()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
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