Many nodal domains in random regular graphs

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.

18 pages. Minor changes to the introduction

References in corpus (5)

• Eigenvectors of the discrete Laplacian on regular graphs - a statistical approach
• Gaussian Waves on the Regular Tree
• Nodal domains on graphs - How to count them and why?
• Delocalization of eigenvectors of random matrices. Lecture notes
• Spectrum of Random $d$-regular Graphs Up to the Edge