Many nodal domains in random regular graphs

Shirshendu Ganguly, Theo McKenzie, Sidhanth Mohanty, Nikhil Srivastava

Published 2021-09-23Version 1

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.