Hodge Laplacians on graphs

TL;DR

This paper provides an accessible introduction to Hodge Laplacians on graphs, emphasizing their linear algebraic properties and topological aspects, making advanced concepts more approachable for those familiar with basic graph theory and linear algebra.

Contribution

It offers a simplified, graph-centric presentation of Hodge Laplacians, highlighting their algebraic and topological properties without requiring complex topological background.

Findings

Hodge Laplacian properties are explained using basic linear algebra.

Cohomology and Hodge theory are shown to be largely algebraic in nature.

The approach simplifies understanding of topological features in graphs.

Abstract

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 $AB = 0$. For the remaining topological aspect, we cast our discussions entirely in terms of graphs as opposed to less-familiar topological objects like simplicial complexes.