论文标题
极端图形实现和图形拉普拉斯特征值
Extremal graph realizations and graph Laplacian eigenvalues
论文作者
论文摘要
对于以原点为中心的常规多面体（或多边形），顶点的坐标是图形laplacian的特征向量，用于该多面体（或多边形）与第一个（非琐事）特征值相关的骨架。在本文中，我们概括了这种关系。对于给定的图，我们研究了特征值优化的问题，即在非负边缘重量上最大化图形拉普拉斯的第一个特征值。我们表明，在某些假设下，使用与该问题解决方案对应的特征向量的图形实现的光谱实现是一个居中的，单位距离图实现，具有最大的总方差。该结果提供了一种生成单位距离图实现的新方法，并基于凸双重性。该方法的缺点是实现的维度是由极端特征值的多样性给出的，极端特征值的多样性在解决特征值优化问题之前通常是未知的。我们的结果用许多示例说明了。
For a regular polyhedron (or polygon) centered at the origin, the coordinates of the vertices are eigenvectors of the graph Laplacian for the skeleton of that polyhedron (or polygon) associated with the first (non-trivial) eigenvalue. In this paper, we generalize this relationship. For a given graph, we study the eigenvalue optimization problem of maximizing the first (non-trivial) eigenvalue of the graph Laplacian over non-negative edge weights. We show that the spectral realization of the graph using the eigenvectors corresponding to the solution of this problem, under certain assumptions, is a centered, unit-distance graph realization that has maximal total variance. This result gives a new method for generating unit-distance graph realizations and is based on convex duality. A drawback of this method is that the dimension of the realization is given by the multiplicity of the extremal eigenvalue, which is typically unknown prior to solving the eigenvalue optimization problem. Our results are illustrated with a number of examples.
