Nodal geometry of graphs on surfaces

TL;DR

This paper extends the Discrete Nodal Theorem to graphs on surfaces, providing bounds on eigenvalue multiplicities for graphs with certain connectivity and genus, applicable to Schrödinger operators.

Contribution

It introduces two mixed versions of the Discrete Nodal Theorem for graphs on surfaces and derives bounds on eigenvalue multiplicities considering graph genus and connectivity.

Findings

Bound on eigenvalue multiplicity for three-connected graphs with volume-growth conditions

Extension of Nodal Theorem to Schrödinger operators on graphs

Results applicable to graphs embedded on surfaces of fixed genus

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 Schr\"odinger operator, not just the Laplacian.