“On the graph Laplacian and its applications in machine learning – Part 1: spectral graph theory”  by Gabriel Chen PhD, Mathematics & Statistics Department

The Laplacian matrix of a graph is an important concept in machine learning. Let G be an undirected, weighted graph with weight matrix W (e.g., a social network in which weights indicate how well two people know each other). The Laplacian matrix of the graph is defined as L = D - W, where D is a diagonal matrix with diagonal entries being the row sums of W. The graph Laplacian matrix (L) has many interesting properties. For example, it is symmetric and positive semidefinite, and the multiplicity of the eigenvalue 0 coincides with the number of connected components in the graph. This is the first of a two-talk series, where we will introduce the concepts associated to the Laplacian matrix in a graph setting and then present some theoretical properties as well as interpretations of the graph Laplacian matrix. The talk is most accessible to people who have taken one semester of linear algebra; knowledge of graph concepts is helpful but not needed.