Eigenvectors of Random Graphs: Nodal Domains

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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 2 nodal domains, together with at most c exceptional vertices falling outside these primary domains. We also discuss variations of these questions and briefly report on some numerical experiments which, in particular, suggest that there are almost surely no exceptional vertices.