论文标题
有限图的第一个拉普拉斯特征值的最大化
Maximization of the first Laplace eigenvalue of a finite graph
论文作者
论文摘要
给定有限图的边缘集的长度函数，我们根据其定义了顶点重量和边缘重量，并考虑相应的图形laplacian。在本文中，我们考虑了在所有边缘长度函数上最大化该laplacian的第一个非零特征值的问题。对于这个问题的极端解决方案，我们证明了从顶点设置到欧几里得空间的地图，该空间由相应的Laplacian的第一个本征函数组成，以便可以按照地图和欧几里得距离明确表示长度函数。这是Nadirashvili的结果的图形 - 与平滑表面上的第一遍历最大化问题有关的结果。我们讨论了简单的示例，也证明了最大化Göring-Helmberg-Wappler问题解决方案的结果。
Given a length function on the edge set of a finite graph, we define a vertex-weight and an edge-weight in terms of it and consider the corresponding graph Laplacian. In this paper, we consider the problem of maximizing the first nonzero eigenvalue of this Laplacian over all edge-length functions subject to a certain normalization. For an extremal solution of this problem, we prove that there exists a map from the vertex set to a Euclidean space consisting of first eigenfunctions of the corresponding Laplacian so that the length function can be explicitly expressed in terms of the map and the Euclidean distance. This is a graph-analogue of Nadirashvili's result related to first-eigenvalue maximization problem on a smooth surface. We discuss simple examples and also prove a similar result for a maximizing solution of the Göring-Helmberg-Wappler problem.
