In the mathematical field of graph theory the Laplacian matrix, sometimes called admittance matrix, Kirchhoff matrix or discrete Laplacian, is a matrix representation of a graph. Together with Kirchhoff's theorem it can be used to calculate the number of spanning trees for a given graph. The Laplacian matrix can be used to find many other properties of the graph; see spectral graph theory. Cheeger's inequality from Riemannian geometry has a discrete analogue involving the Laplacian Matrix; this is perhaps the most important theorem in Spectral Graph theory and one of the most useful facts in algorithmic applications. It approximates the sparsest cut of a graph through the second eigenvalue of its Laplacian.
That is, it is the difference of the degree matrix D and the adjacency matrix A of the graph. In the case of directed graphs, either the indegree or outdegree might be used, depending on the application.