Upper bounds for the Steklov eigenvalues on trees

Monotonicity of Steklov eigenvalues on graphs and applications

“…obtained in another interesting work [7] by them, in a significant way. Here G is also a tree and diam(G) means the diameter of G. To obtain (1.1), He and Hua established a theory of Steklov flows in [8] and as an application of their theory, they also characterized the rigidity of the isodiametric estimate (1.2).…”

“…Then, Remark 4.1. Because a star of degree two and a regular comb with two teeth are both paths, the estimates above for trees containing a star or a regular comb as a subtree can be viewed as generalization of the isodiametric estimate in He-Hua [7].…”

“…There are quite a number of works on exploring the properties Steklov eigenvalues on graphs in recent years. See for example [5,7,8,10,12,13,14,15,16].…”

Steklov flows on trees and applications

“…For subgraphs in Cayley graphs of discrete groups of polynomial growth, the upper bound estimates were proved in [Per20]; see also [HH19]. In our previous paper [HH20], we obtained various upper bounds of Steklov eigenvalues on finite trees. Note that infinite trees are regarded as discrete counterparts of Hadamard manifolds.…”

“…In our previous paper [HH20], we prove the upper bound estimate of the first nonzero Steklov eigenvalue using the diameter of a finite tree; see [HH20,Theorem 1.4]. As an application of the main result, we characterize the equality case for the upper bound estimate; see Theorem 5.2.…”