Comparison between the first Steklov eigenvalue and algebraic connectivity on trees

TL;DR

This paper compares the first Steklov eigenvalue and algebraic connectivity in trees, revealing that extremal trees are the same for both but their eigenvalues differ significantly, highlighting differences in spectral properties.

Contribution

It provides a novel comparison between Steklov eigenvalues and algebraic connectivity in trees with fixed boundary vertices and matching number.

Findings

Extremal trees are identical for both operators.

Steklov eigenvalues and algebraic connectivity differ significantly.

The study highlights spectral property differences in trees.

Abstract

Trees can be regarded as discrete analogue of Hadamard manifolds, namely simply-connected Riemannian manifolds of non-positive sectional curvature. In this paper, we compare the first (non-trivial) Steklov eigenvalue and algebraic connectivity of trees with prescribed number of boundary vertices and matching number. It is particularly noteworthy that while the extremal trees coincide for both operators, their corresponding eigenvalues differ significantly.