广义图拉普拉斯算子的代数性质：电阻网络、临界群和同调代数
Algebraic Properties of Generalized Graph Laplacians: Resistor Networks, Critical Groups, and Homological Algebra
摘要：
我们提出了一个代数框架的广义图拉普拉斯算子，它统一的电阻网络，临界群，拉普拉斯算子和邻接矩阵的特征值的研究。给定一个具有边界$G$的图和一个广义Laplacian $L$，其元素在一个交换环$R$中，我们定义了一个广义临界群$Υ_R(G,L)$ .我们使用同调代数的Hom、Tor和Ext函子将$Υ_R(G,L)$与网络上的调和函数空间联系起来。我们研究了这些代数对象如何在网络$(G,L)$上进行组合运算，包括调和态射、层剥离、对偶和对称。特别是，我们使用层剥离操作的理论电阻网络系统化离散谐波连续。这导致了一个代数特征的图的边界，可以完全层剥离，一个算法，简化计算的$Υ_R(G,L)$，和上界的不变因子的数量在临界组和多重拉普拉斯特征值的几何量。
英文摘要：
We propose an algebraic framework for generalized graph Laplacians which unifies the study of resistor networks, the critical group, and the eigenvalues of the Laplacian and adjacency matrices. Given a graph with boundary $G$ together with a generalized Laplacian $L$ with entries in a commutative ring $R$, we define a generalized critical group $Υ_R(G,L)$. We relate $Υ_R(G,L)$ to spaces of harmonic functions on the network using the Hom, Tor, and Ext functors of homological algebra. We study how these algebraic objects transform under combinatorial operations on the network $(G,L)$, including harmonic morphisms, layer-stripping, duality, and symmetry. In particular, we use layer-stripping operations from the theory of resistor networks to systematize discrete harmonic continuation. This leads to an algebraic characterization of the graphs with boundary that can be completely layer-stripped, an algorithm for simplifying computation of $Υ_R(G,L)$, and upper bounds for the number of invariant factors in the critical group and the multiplicity of Laplacian eigenvalues in terms of geometric quantities.
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