Hodge Laplacians on graphs

This is an elementary introduction to the Hodge Laplacian on a graph, a higher-order generalization of the graph Laplacian. We will discuss basic properties including cohomology and Hodge theory. At the end we will also discuss the nonlinear Laplacian on a graph, a nonlinear generalization of the graph Laplacian as its name implies. These generalized Laplacians will be constructed out of coboundary operators, i.e., discrete analogues of exterior derivatives. The main feature of our approach is simplicity — this article requires only knowledge of linear algebra and graph theory.