Eigenvectors of random graphs: Nodal Domains

Yael Dekel, James R. Lee*, Nathan Linial

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

30 Scopus citations

Abstract

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 article 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.

Original languageAmerican English
Pages (from-to)39-58
Number of pages20
JournalRandom Structures and Algorithms
Volume39
Issue number1
DOIs
StatePublished - Aug 2011

Keywords

  • Eigenvectors
  • Random graphs
  • Spectral analysis

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