Abstract

An eigenfunction of the Laplacian on a metric (quantum) graph has an excess number of zeros due to the graph’s non-trivial topology. This number, called the nodal surplus, is an integer between 0 and the graph’s first Betti number β. We study the distribution of the nodal surplus values in the countably infinite set of the graph’s eigenfunctions. We conjecture that this distribution converges to Gaussian for any sequence of graphs of growing β. We prove this conjecture for several special graph sequences and test it numerically for a variety of well-known graph families. Accurate computation of the distribution is made possible by a formula expressing the nodal surplus distribution as an integral over a high-dimensional torus.

摘要

由于图的非平凡拓扑，度量（量子）图上的拉普拉斯算子的特征函数具有过多的零。这个数字称为节点剩余，是一个介于 0 和图的第一个 Betti 数β之间的整数。我们研究了图的特征函数的可数无限集中节点剩余值的分布。我们推测，对于任何增长β的图序列，该分布都会收敛到高斯分布。我们针对几个特殊的图序列证明了这个猜想，并针对各种著名的图族对其进行了数值测试。通过将节点剩余分布表示为高维环面上的积分的公式，可以准确计算分布。