Many important properties of an undirected graph manifest themselves spectrally in the eigenvalues or quadratic forms of matrices related to the graph. For instance, the connectivity structure, electrical properties, and random walk behavior of a graph are determined by its Laplacian matrix. A spectral sparsifier of a graph G is a sparse graph H on the same set of vertices such that the Laplacians of H and G are close, so that H captures the spectral behavior of G while being much cheaper to store and perform computations on. We survey a line of work showing that spectral sparsifiers with constant degree exist for every graph and can be computed efficiently.
Graphs Across Domains (GraphXD) is a BIDS working group that connects scientists, researchers, and theorists interested in graphs (networks) from a variety of fields (including mathematics, computer science, biology, physics, economics, and sociology) through seminars and workshops.