 
Summary: Co~mr~^roRicA 6 (2) (1986) 8396
EIGENVALUES AND EXPANDERS
N. ALON
Received31 January1985
Revised10 September1985
Linear expanders have numerous applications to theoretical computer science. Here we show
that a regular bipartite graphis an expander ifandonlyifthe second largest eigenvalue ofits adjacency
matrix is wellseparated from the first. This result, which has an analytic analogue for Riemannian
manifolds enables one to generate expanders randomly and check efficientlytheir expanding proper
ties. It also supplies an efficient algorithm for approximating the expanding properties of a graph.
The exact determination of these properties is known to be coNPcomplete.
1. Introduction
Let G = (V, E) be a graph. For a subset X of V put
N(X) = {vCV: vxCE for some x~X}.
An (n, d, c)expander is a bipartite graph on the sets of vertices 1 (inputs) and O (out
puts), where [I[=[Ol=n, the maximal degree of a vertex is d, and for every set
XC=l of cardinality IXl=~<n/2,
(1.1) [N(X)I ~ (1 +c(l c~/n)). ct.
It is a strong (n, d, c)expander if (1.1) holds for all XC=I. A family of linear expand
ers of density d and expansion c is a set {Gi}?=l, where Gi is an (hi, d, c)expander,
