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 $n$th Laplacian eigenvalue is at most $2{\left[6(n-1)+15(2g-2)\right]}^2$. Our results hold for any Schrödinger operator, not just the Laplacian.