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

Let $T$ be a finite tree with leaf set $\dO$ as the boundary and let $\lambda_2$ be the first nontrivial Steklov eigenvalue. Let $D$ and $\ell$ be the maximum vertex degree and the number of leaves, respectively. Motivated by the spectral influence of neck-stretching on Riemannian manifolds, we investigate a discrete counterpart--edge-stretching--and its effect on the Steklov eigenvalues of graphs. We prove that Steklov eigenvalues decrease monotonically under the edge--stretching operation. As a consequence, we prove that $\lambda_2\le D/\ell$, with equality if and only if $T$ is a star. This fundamentally improves the constant in He--Hua's bound $\lambda_2\le 4(D-1)/\ell$ to the optimal value~$1$. We also provide a closed-form diagonalization of the Steklov problem on level--regular trees, yielding explicit eigenvalues and multiplicities. In addition, we provide a general upper bound $\lambda_k\le \min\{1,\,16Dk/\ell\}$ for higher eigenvalues. Systematic numerical experiments verify the sharp bound and provide evidence for the extremal conjecture of Lin--Zhao on balanced minimum--height trees.