Let G be a random d-regular graph on n vertices. We prove that for every constant α > 0 \documentclass[12pt]{minimal}
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\begin{document}$$\alpha > 0$$\end{document} , with high probability every eigenvector of
the adjacency matrix of G with eigenvalue less than - 2 d - 2 - α \documentclass[12pt]{minimal}
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… • Our question: can we prove nontrivial structure of nodal domains for a randomly
selected 𝑑𝑑-regular graph, for fixed 𝑑𝑑. … • This is the first result lower bounding the
number of nodal domains for random regular graphs. … • We expect there to be many nodal
domains of small size, and then two large nodal domains that contain almost all vertices. …