Algorithms, Graph Theory, and Linear Equations in Laplacian Matrices

The Laplacian matrices of graphs are fundamental. In addition to facilitating the application of linear algebra to graph theory, they arise in many practical problems. In this talk we survey recent progress on the design of provably fast algorithms for solving linear equations in the Laplacian matrices of graphs. These algorithms motivate and rely upon fascinating primitives in graph theory, including low-stretch spanning trees, graph sparsifiers, ultra-sparsifiers, and local graph clustering. These are all connected by a definition of what it means for one graph to approximate another. While this definition is dictated by Numerical Linear Algebra, it proves useful and natural from a graph theoretic perspective. Mathematics Subject Classification (2010). Primary 68Q25; Secondary 65F08.