4.6 Article

Many Nodal Domains in Random Regular Graphs

Journal

COMMUNICATIONS IN MATHEMATICAL PHYSICS
Volume 401, Issue 2, Pages 1291-1309

Publisher

SPRINGER
DOI: 10.1007/s00220-023-04709-6

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For a random d-regular graph G on n vertices, we prove that with high probability, every eigenvector of G's adjacency matrix with eigenvalue less than -2 root d - 2 - alpha has at least Omega (n/polylog(n)) nodal domains.
Let G be a random d-regular graph on n vertices. We prove that for every constant alpha > 0, with high probability every eigenvector of the adjacency matrix of G with eigenvalue less than -2 root d - 2 -alpha a has Omega (n/polylog(n)) nodal domains.

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