  We initiate a systematic study of eigenvectors of random graphs. Whereas much
is known about eigenvalues of graphs and how they reflect properties of the
underlying graph, relatively little is known about the corresponding
eigenvectors. Our main focus in this paper is on the nodal domains associated
with the different eigenfunctions. In the analogous realm of Laplacians of
Riemannian manifolds, nodal domains have been the subject of intensive research
for well over a hundred years. Graphical nodal domains turn out to have
interesting and unexpected properties. Our main theorem asserts that there is a
constant c such that for almost every graph G, each eigenfunction of G has at
most two large nodal domains, and in addition at most c exceptional vertices
outside these primary domains. We also discuss variations of these questions
and briefly report on some numerical experiments which, in particular, suggest
that almost surely there are just two nodal domains and no exceptional
vertices.
