Nodal geometry of graphs on surfaces

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 $2\left[ 6(n-1) + 15(2g-2) \right]^2$. Our results hold for any Schr\"odinger operator, not just the Laplacian.