Hodge Laplacians on graphs

Understand

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.
• The main feature of our approach is simplicity, requiring only knowledge of linear algebra and graph theory.
• We have also isolated the algebra from the topology to show that a large part of cohomology and Hodge theory is nothing more than the linear algebra of matrices satisfying $AB = 0$.