Nodal geometry of graphs on surfaces
Yong Lin Gábor Lippner Dan Mangoubi Shing-Tung Yau
We prove two mixed versions of the Discrete Nodal Theorem of Davies et. al. [3] for bounded degree graphs, and for three-connected graphs of fixed genus $g$. Using this we can show that for a three-connected graph satisfying a certain volume-growth condition, the multiplicity of the $n$th Laplacian eigenvalue is at most $2[ 6(n-1) + 15(2g-2)]^2$. Our results hold for any Schrödinger operator, not just the Laplacian.
keywords: genus.test multiplicity of eigenvalues Nodal domain

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