Nodal geometry of graphs on surfaces

Abstract

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 nth Laplacian eigenvalue is at most 2 [6(n - 1) + 15(2g - 2)]2. Our results hold for any Schrodinger operator, not just the Laplacian.