SIAM Review, Volume 62, Issue 3, Page 683-683, January 2020.
This issue of SIAM Review contains two papers in the Education section. The first paper, “Hodge Laplacians on Graphs,” is presented by Lek-Heng Lim. The classical Hodge Laplacian is a differential operator defined on any manifold equipped with a Riemannian metric. In this paper, the author provides an accessible introduction to what he calls graph-theoretic Hodge theory. The Hodge Laplacian on a graph is a higher-order generalization of the graph Laplacian. The reader needs only knowledge of linear algebra and graph theory in order to follow the exposition; however, much broader knowledge in mathematics would help one fully understand and appreciate this theory. The author's approach is to derive a large part of cohomology theory in finite dimensions by analyzing the simple structure of $AB=0$ for two matrices $A$ and $B$. The Hodge Laplacian is the matrix $A^{\ast} A + BB^{\ast}$. The topological aspects involved are presented entirely in the terminology of graphs. The paper contains a discussion of fundamental concepts such as Hodge decomposition, harmonic vector fields, and coboundary operators of first and higher order. The author introduces real Euclidean vector spaces of totally antisymmetric (alternating) functions on ordered sets of $k$-cliques ($k$-cochains) and shows that the permutation of arguments of an alternating function changes its value by the sign of the permutation. He defines notions like combinatorial gradient and combinatorial curl, which become instrumental in developing Helmholtz-type decomposition for graphs. The paper contains discrete analogues to statements about smooth vector fields, such as “a vector field is curl-free and divergence-free if and only if it is a harmonic vector field.” Some attention is paid to computing the quantities appearing in the theoretical analysis and to modern data applications of this theory. This ties in neatly with our recent Research Spotlights article on “Random Walks on Simplicial Complexes and the Normalized Hodge 1-Laplacian,” by Michael T. Schaub, Austin R. Benson, Paul Horn, Gabor Lippner, and Ali Jadbabaie (SIAM Review, 62 (2) (2020), pp. 353--391). The second paper presents “An Elementary Proof of a Matrix Tree Theorem for Directed Graphs,” written by Patrick De Leenheer. In graph theory, Kirchhoff's matrix tree theorem, named after Gustav Kirchhoff, establishes the number of spanning trees in a graph. For a connected undirected graph, the number of spanning trees, rooted at any vertex of it, is equal to the determinant of the reduced Laplacian matrix associated with the graph. Evidently, this number can be computed in polynomial time. The result has been generalized by W. T. Tutte in 1948 to the case of directed graphs. Given a directed graph with labeled vertices, we can define two Laplacians: $L_1$ is the difference of the in-degree matrix minus the adjacency matrix of the graph, while $L_2$ is the difference of the out-degree matrix and the transposed adjacency matrix. For a fixed vertex $v_r$, two reduced Laplacians $L_1^r$ and $L_2^r$ are obtained by removing the $r$th row and the $r$th column from $L_1$ and $L_2$, respectively. The Tutte's theorem states that the numbers of outgoing and incoming directed spanning trees rooted at $v_r$ are equal to the determinant of $L_1^r$ and $L_2^r$, respectively. The author provides a proof of this result. Additional observation relates the number of the spanning trees to the eigenvectors of the two Laplacians, corresponding to the eigenvalue zero. The results generalize to weighted graphs.

中文翻译：

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教育

SIAM评论，第62卷，第3期，第683-683页，2020年1月。
本期《 SIAM评论》在“教育”部分包含两篇论文。Lek-Heng Lim提出了第一篇论文“图上的Hodge Laplacians”。经典的Hodge Laplacian是在任何配备黎曼度量的歧管上定义的微分算子。在本文中，作者提供了他所谓的图论霍奇理论的易于理解的介绍。图上的Hodge Laplacian是图Laplacian的高阶概括。读者只需要了解线性代数和图论的知识，就可以理解本讲解。但是，更广泛的数学知识将有助于人们充分理解和理解这一理论。作者的方法是通过分析两个矩阵$ A $和$ B $的$ AB = 0 $的简单结构，在有限维中推导大部分同调理论。霍奇拉普拉斯算子是矩阵$ A ^ {\ ast} A + BB ^ {\ ast} $。涉及的拓扑方面完全以图形术语表示。本文包含有关基本概念的讨论，例如Hodge分解，谐波矢量场以及一阶和更高阶的共边界算子。作者在$ k $ -cliques（$ k $ -cochains）的有序集上引入了完全反对称（交替）函数的实欧氏向量空间，并显示了交替函数的参数的排列通过排列的符号改变了其值。他定义了诸如组合梯度和组合卷曲之类的概念，这些概念在开发图的亥姆霍兹类型分解时起了重要作用。本文包含有关平滑向量场的陈述的离散类似物，例如““当且仅当它是谐波向量场时，向量场才是无卷曲和无散度的。” 在计算理论分析中出现的数量以及该理论的现代数据应用方面给予了一定的关注。这与我们最近的研究重点摘要文章“ Michael R. Schaub，Austin R. Benson，Paul Horn，Gabor Lippner和Ali Jadbabaie（《 SIAM评论》上的“随机行走在简单复合体和标准化Hodge 1-Laplacian上”）紧密相关。 62（2）（2020），第353--391页）。第二篇论文提出了“有向图的矩阵树定理的基本证明”，由Patrick De Leenheer撰写。在图论中，以古斯塔夫·基希霍夫（Gustav Kirchhoff）命名的基希霍夫矩阵树定理确定了图中的生成树数。对于连接的无向图，生成树的数量，根于其任何顶点的根等于与该图关联的简化拉普拉斯矩阵的行列式。显然，这个数字可以用多项式时间来计算。WT Tutte在1948年将结果推广到有向图的情况。给定带有标记顶点的有向图，我们可以定义两个拉普拉斯算子：$ L_1 $是度矩阵的差减去图的邻接矩阵，而$ L_2 $是度矩阵和转置矩阵的差邻接矩阵。对于固定顶点$ v_r $，通过分别从$ L_1 $和$ L_2 $中删除$ r $ th行和$ r $ th列，可以获得两个简化的拉普拉斯算子$ L_1 ^ r $和$ L_2 ^ r $。图特定理指出，以$ v_r $为根的传出和传入有向生成树的数量等于$ L_1 ^ r $和$ L_2 ^ r $的行列式，分别。作者提供了这一结果的证明。附加观察将生成树的数量与两个Laplacian的特征向量相关联，对应于特征值零。结果概括为加权图。