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DCDS
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.
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