Upper bounds for the Steklov eigenvalues on trees

TL;DR

This paper establishes sharp upper bounds for discrete Steklov eigenvalues on trees, relating them to geometric properties like boundary size and diameter, and extends these bounds to higher eigenvalues.

Contribution

The paper introduces new sharp upper bounds for Steklov eigenvalues on trees based on geometric quantities, including for higher order eigenvalues.

Findings

Sharp upper bounds for the first nonzero Steklov eigenvalue based on boundary size and diameter.

Extension of bounds to higher order Steklov eigenvalues.

Results are applicable to finite trees with geometric considerations.

Abstract

In this paper, we study the upper bounds for discrete Steklov eigenvalues on trees via geometric quantities. For a finite tree, we prove sharp upper bounds for the first nonzero Steklov eigenvalue by the reciprocal of the size of the boundary and the diameter respectively. We also prove similar estimates for higher order Steklov eigenvalues.