Cohomology of digraphs and (undirected) graphs
Cohomology of digraphs and (undirected) graphs
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DOI：
10.4310/ajm.2015.v19.n5.a5
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发表时间：
2015
影响因子：
0.6
通讯作者：
A. Grigor’yan;Yong Lin;Y. Muranov;S. Yau
中科院分区：
文献类型：
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作者：
A. Grigor’yan;Yong Lin;Y. Muranov;S. Yau
基于有向图顶点上的函数代数的泛演算，构造了一类有限有向图（有向图）的上同调理论。我们开发必要的代数技术，并将其应用于调查函的性质，这一理论。我们引入类别的有向图和（无向）图，并利用自然同构的引进类的图和对称有向图的全子类，我们转移到我们的上同调理论的类的图。然后证明了无向图的上同调理论的同伦不变性。这样我们就回答了Babson，Jeso，隆盖维尔和Laubenbacher关于图的同伦不变同调理论存在性的问题.我们建立连接自然产生的一些特殊类别的有向图的单纯复形的上同调。例如，偏序集的上同调和与偏序集相关的单纯复形的上同调一致。然而，一般情况下，有向图上同调理论不能归结为单纯上同调。我们描述了有向图上同调群在有向图层次上的几种拓扑结构的行为，证明了任意给定的有限非负整数序列都可以实现为有向图上同调群的秩序列.我们也提出了足够多的例子来说明理论。
We construct a cohomology theory on a category of finite digraphs (directed graphs), which is based on the universal calculus on the algebra of functions on the vertices of the digraph. We develop necessary algebraic technique and apply it for investigation of functorial properties of this theory. We introduce categories of digraphs and (undirected) graphs, and using natural isomorphism between the introduced category of graphs and the full subcategory of symmetric digraphs we transfer our cohomology theory to the category of graphs. Then we prove homotopy invariance of the introduced cohomology theory for undirected graphs. Thus we answer the question of Babson, Barcelo, Longueville, and Laubenbacher about existence of homotopy invariant homology theory for graphs. We establish connections with cohomology of simplicial complexes that arise naturally for some special classes of digraphs. For example, the cohomologies of posets coincide with the cohomologies of a simplicial complex associated with the poset. However, in general the digraph cohomology theory can not be reduced to simplicial cohomology. We describe the behavior of digraph cohomology groups for several topological constructions on the digraph level and prove that any given finite sequence of non-negative integers can be realized as the sequence of ranks of digraph cohomology groups. We present also sufficiently many examples that illustrate the theory.