Maximize the Steklov eigenvalue of trees

We study the maximal Steklov eigenvalues of trees with given number of boundary vertices and total number of vertices. Trees can be regarded as discrete analogue of Hadamard manifolds, namely simply connected Riemannian manifolds of nonpositive sectional curvature. Let be the maximal of th Steklov eigenvalue of trees with leaves as boundary and vertices. We determine that and we characterize the trees attaining this bound. For , we show that . We also give a lower bound on the maximal Steklov eigenvalues of trees with given diameter and total number of vertices. Our work can be regarded as a completion of the work by He–Hua ( Calc. Var. Partial Differential Equations 61 (2022), 101) and Yu–Yu ( Calc. Var. Partial Differential Equations 63 (2024), 22).