Quantum trees which maximize higher eigenvalues are unbalanced

TL;DR

This paper investigates the problem of maximizing higher Laplacian eigenvalues on metric trees, revealing that unbalanced star graphs optimize these eigenvalues, contrasting with the balanced shapes that maximize the first eigenvalue.

Contribution

It identifies the unique unbalanced star graph configurations that maximize higher eigenvalues, extending understanding of eigenvalue optimization on metric trees.

Findings

The k-th eigenvalue is maximized by a star graph with specific unbalanced edge lengths.

The first eigenvalue is maximized by equilateral star graphs.

Higher eigenvalue optimizers tend to be less balanced in shape.

Abstract

The isoperimetric problem of maximizing all eigenvalues of the Laplacian on a metric tree graph within the class of trees of a given average edge length is studied. It turns out that, up to rescaling, the unique maximizer of the $k$-th positive eigenvalue is the star graph with three edges of lengths $2 k - 1$, $1$ and $1$. This complements the previously known result that the first nonzero eigenvalue is maximized by all equilateral star graphs and indicates that optimizers of isoperimetric problems for higher eigenvalues may be less balanced in their shape -- an observation which is known from numerical results on the optimization of higher eigenvalues of Laplacians on Euclidean domains.