Spectrahedral Geometry of Graph Sparsifiers

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.