Size of nodal domains of the eigenvectors of a G(n,p) graph

TL;DR

This paper investigates the sizes of nodal domains of eigenvectors in G(n,p) graphs, showing that, with high probability, these domains are approximately equal in size, revealing symmetry in their structure.

Contribution

The paper proves that in G(n,p) graphs, the two nodal domains of eigenvectors are nearly equal in size with high probability, providing new insights into eigenvector structure.

Findings

Nodal domains are typically of similar size.

High probability results for eigenvector nodal domains.

Symmetry in eigenvector nodal domain sizes.

Abstract

Consider an eigenvector of the adjacency matrix of a G(n, p) graph. A nodal domain is a connected component of the set of vertices where this eigenvector has a constant sign. It is known that with high probability, there are exactly two nodal domains for each eigenvector corresponding to a non-leading eigenvalue. We prove that with high probability, the sizes of these nodal domains are approximately equal to each other.