On the hot spots of quantum trees

TL;DR

This paper investigates the extremal properties of second eigenfunctions of the Laplacian on metric trees, revealing they attain extremal values only at leaf vertices, with examples showing these do not always correspond to the graph's diameter.

Contribution

It establishes a new extremal property of second Laplacian eigenfunctions on metric trees and provides counterexamples related to the graph's diameter.

Findings

Second eigenfunctions attain extremal values only at degree one vertices.

Counterexample where extremal vertices do not realize the diameter.

Provides insight into eigenfunction behavior on metric trees.

Abstract

We show that any second eigenfunction of the Laplacian with standard vertex conditions on a metric tree graph attains its extremal values only at degree one vertices, and give an example where these vertices do not realise the diameter of the graph.