Eigenvectors of random graphs: Nodal domains

TL;DR

This paper explores the properties of eigenvectors of random graphs, revealing that most eigenfunctions have at most two large nodal domains and few exceptional vertices, with numerical evidence supporting these findings.

Contribution

It introduces a systematic study of eigenvector nodal domains in random graphs and establishes bounds on their number, a novel insight compared to prior eigenvalue-focused research.

Findings

Most eigenfunctions have at most two large nodal domains

There are at most c exceptional vertices outside the primary domains

Numerical experiments suggest almost surely only two nodal domains and no exceptional vertices

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 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.