Nodal Statistics On Quantum Graphs

TL;DR

This paper investigates the distribution of nodal surpluses of Laplacian eigenfunctions on metric graphs, revealing symmetry properties and a universal binomial distribution for graphs with disjoint cycles, linking spectral data to graph topology.

Contribution

It establishes the existence and symmetry of the nodal surplus distribution for metric graphs and proves a universal binomial form for graphs with disjoint cycles, introducing local surpluses and their independence.

Findings

Distribution of nodal surplus exists and is symmetric.

First Betti number equals twice the average nodal surplus.

For disjoint cycles, the distribution is binomial, with local surpluses behaving like independent Bernoulli variables.

Abstract

It has been suggested that the distribution of the suitably normalized number of zeros of Laplacian eigenfunctions contains information about the geometry of the underlying domain. We study this distribution (more precisely, the distribution of the "nodal surplus") for Laplacian eigenfunctions of a metric graph. The existence of the distribution is established, along with its symmetry. One consequence of the symmetry is that the graph's first Betti number can be recovered as twice the average nodal surplus of its eigenfunctions. Furthermore, for graphs with disjoint cycles it is proven that the distribution has a universal form --- it is binomial over the allowed range of values of the surplus. To prove the latter result, we introduce the notion of a local nodal surplus and study its symmetry and dependence properties, establishing that the local nodal surpluses of disjoint cycles behave like independent Bernoulli variables.