SIAM Review, Volume 62, Issue 3, Page 685-715, January 2020.
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 discussion entirely in terms of graphs as opposed to less familiar topological objects like simplicial complexes.

中文翻译：

---

图上的Hodge Laplacians

SIAM评论，第62卷，第3期，第685-715页，2020年1月。
这是对图上Hodge Laplacian的基本介绍，是图Laplacian图的高阶概括。我们将讨论基本特性，包括同调性和霍奇理论。我们方法的主要特点是简单，只需要线性代数和图论知识。我们还从拓扑中分离了代数，以表明大部分的同调学和Hodge理论仅是满足$ AB = 0 $的矩阵的线性代数。对于其余的拓扑方面，我们将讨论完全从图的角度进行，而不是像简单复形这样不太熟悉的拓扑对象。