Effect of edge-stretching on Steklov eigenvalues and sharp Steklov eigenvalue bounds on leaf–boundary trees

Abstract

Let T be a finite tree with leaf set as the boundary and let λ₂ be the first nontrivial Steklov eigenvalue. Motivated by neck-stretching on Riemannian manifolds, we study a discrete counterpart—edge-stretching—and prove that Steklov eigenvalues decrease monotonically under this operation. As a consequence, λ₂ ≤ D/ℓ, with equality if and only if T is a star, improving the constant in He–Hua’s bound to the optimal value 1. We also give a closed-form diagonalization on level-regular trees, an upper bound for higher eigenvalues, and numerical evidence for the Lin–Zhao extremal conjecture.