We propose an approach to graph sparsification based on the idea of preserving the smallest $k$ eigenvalues and eigenvectors of the Graph Laplacian. This is motivated by the fact that small eigenvalues and their associated eigenvectors tend to be more informative of the global structure and geometry of the graph than larger eigenvalues and their eigenvectors. The set of all weighted subgraphs of a graph $G$ that have the same first $k$ eigenvalues (and eigenvectors) as $G$ is the intersection of a polyhedron with a cone of positive semidefinite matrices. We discuss the geometry of these sets and deduce the natural scale of $k$. Various families of graphs illustrate our construction.

翻译：我们提出一种基于保留图拉普拉斯矩阵最小$k$个特征值及特征向量思想的图谱稀疏化方法。其动机在于：相较于较大特征值及其特征向量，较小特征值及其关联特征向量通常更能反映图的整体结构与几何特性。对于图$G$的所有加权子图，若其与$G$共享前$k$个特征值（及特征向量），则这些子图构成的集合可表示为多面体与半正定矩阵锥的交集。本文讨论这些集合的几何性质，并推导出$k的自然尺度。多种图族实例说明了我们的构造方法。