Extremal graph realizations and graph Laplacian eigenvalues

B. Osting
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Abstract

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.
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极值图实现和图拉普拉斯特征值
对于以原点为中心的正多面体(或多边形),顶点的坐标是与第一个(非平凡)特征值相关的多面体(或多边形)骨架的图拉普拉斯特征向量。在本文中,我们推广了这一关系。对于给定的图,我们研究了在非负边权的拉普拉斯算子上最大化图的第一个(非平凡)特征值的特征值优化问题。我们证明,在一定的假设下,使用与该问题的解相对应的特征向量的图的谱实现是具有最大总方差的有中心的、单位距离的图实现。该结果给出了一种基于凸对偶的单位距离图实现的新方法。该方法的一个缺点是实现的维度是由极值特征值的多重性给出的,而极值特征值在求解特征值优化问题之前通常是未知的。我们的结果用一些例子来说明。
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