2026/07/22 by Lixiang Chen, Bobo Hua, Yongtang Shi
#math.CO #math.SP
Let G=(V,E) be a finite connected graph with boundary B. We prove that for a generic positive edge weight function w ∈ 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.