Steklov flows on trees and applications

Abstract

We introduce the Steklov flows on finite trees, i.e. the flows (or currents) associated with the Steklov problem. By constructing appropriate Steklov flows, we prove the monotonicity of the first nonzero Steklov eigenvalues on trees: for finite trees G1 and G2, the first nonzero Steklov eigenvalue of G1 is greater than or equal to that of G2, provided that G1 is a subgraph of G2.

Monotonicity of Steklov eigenvalues on graphs and applications

“…This means that the assumption (1.26) in Theorem 1.1 will be automatically satisfied on trees with unit weight. Therefore, we have the following corollary for trees with unit weight which gives an affirmative answer to Problem 1.5 in [8].…”

“…In fact, we obtained monotonicity of Steklov eigenvalues for general weighted graphs. Our method is different with that of He and Hua in [8].…”

“…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].…”

“…In a recent interesting work [8], He and Hua obtained a monotonicity of the first positive Steklov eigenvalues on trees equipped with the unit weight. More precisely, they showed that…”

“…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). In their work [8], He and Hua also proposed two interesting problems about extending the monotonicity (1.1) to higher Steklov eigenvalues and to general graphs.…”