Generic properties of discrete Steklov eigenfunctions

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.