Generic properties of discrete Steklov eigenfunctions

Abstract

Let $G=(V,E)$ be a finite connected graph with boundary $B$. We prove that for a generic positive edge weight function $w\in {\mathbb{R}}^{|E|}$, the Steklov eigenvalues of $(G,B,w)$ are simple and every Steklov eigenfunction does not vanish on the boundary. More precisely, the exceptional weights are contained in a zero set of a non-identically zero polynomial and hence form a set of Lebesgue measure zero and Hausdorff dimension at most $|E|-1$. Our results provide a discrete extension of the genericity theorem for the Steklov problem on compact manifolds.

16 pages