Graph structure of the nodal set and bounds on the number of critical points of eigenfunctions on Riemannian manifolds

Abstract

In this article, we illustrate and draw connections between the geometry of zero sets of eigenfunctions, graph theory and the vanishing order of eigenfunctions. We identify the nodal set of an eigenfunction of the Laplacian (with smooth potential) on a compact, two-dmensional Riemannian manifolds, that is on Riemannian surfaces, as an embedded metric graph and then use tools from elementary graph theory in order to estimate the number of critical points in the nodal set of the $k$-th eigenfunction and the sum of vanishing orders at critical points in terms of $k$ and the Euler-Poincar\'e characteristic of the surface.