Many nodal domains in random regular graphs

TL;DR

This paper proves that in random regular graphs, eigenvectors with eigenvalues below a certain threshold typically have a large number of nodal domains, highlighting complex eigenvector structures.

Contribution

It establishes a high-probability lower bound on the number of nodal domains for eigenvectors with sufficiently low eigenvalues in random regular graphs.

Findings

Eigenvectors with eigenvalues less than -2√(d-2)-α have many nodal domains.

High probability results for the structure of eigenvectors in random regular graphs.

Quantitative bounds on the number of nodal domains in terms of graph size.

Abstract

Let $G$ be a random $d$-regular graph. We prove that for every constant $\alpha > 0$, with high probability every eigenvector of the adjacency matrix of $G$ with eigenvalue less than $-2\sqrt{d-2}-\alpha$ has $\Omega(n/$polylog$(n))$ nodal domains.