Eigenvalue estimates for the Laplacian on a metric tree

TL;DR

This paper derives explicit upper bounds for Laplacian eigenvalues on finite metric trees, including a sharp bound for the spectral gap, with equilateral star graphs uniquely maximizing it among trees of fixed average length.

Contribution

It provides new explicit bounds for Laplacian eigenvalues on metric trees and characterizes the maximizers of the spectral gap.

Findings

Sharp upper bound for the spectral gap on metric trees

Equilateral star graphs uniquely maximize the spectral gap

Eigenvalue estimates depend on average edge length or diameter

Abstract

We provide explicit upper bounds for the eigenvalues of the Laplacian on a finite metric tree subject to standard vertex conditions. The results include estimates depending on the average length of the edges or the diameter. In particular, we establish a sharp upper bound for the spectral gap, i.e. the smallest positive eigenvalue, and show that equilateral star graphs are the unique maximizers of the spectral gap among all trees of a given average length.