论文标题
关于独立综合体的同源
On the homology of independence complexes
论文作者
论文摘要
图$ g $的独立复合物$ \ mathrm {ind}（g）$是由其独立集形成的简单复合物。本文介绍了$ \ mathrm {indrm {ind}（g）$的简单边界图的变形，该图引起了带有琐碎同源性的双重复合体。朝着正确方向过滤这个双重复合体会诱导光谱序列，该频谱序列收敛到零，并在其第一页上包含$ g $的独立综合体的同源性和$ g $的各种子学，通过从$ g $中删除独立的集合及其社区获得。结果表明，该光谱序列可用于研究$ \ mathrm {ind}（g）$的同源性。此外，对序列第一页的仔细调查表明，$ g $中最大独立集的基数与某些$ G $的某些独立综合体的某些同源群的消失之间的关系。该关系显示为所有路径和环状图。
The independence complex $\mathrm{Ind}(G)$ of a graph $G$ is the simplicial complex formed by its independent sets. This article introduces a deformation of the simplicial boundary map of $\mathrm{Ind}(G)$ that gives rise to a double complex with trivial homology. Filtering this double complex in the right direction induces a spectral sequence that converges to zero and contains on its first page the homology of the independence complexes of $G$ and various subgraphs of $G$, obtained by removing independent sets and their neighborhoods from $G$. It is shown that this spectral sequence may be used to study the homology of $\mathrm{Ind}(G)$. Furthermore, a careful investigation of the sequence's first page exhibits a relation between the cardinality of maximal independent sets in $G$ and the vanishing of certain homology groups of the independence complexes of some subgraphs of $G$. This relation is shown to hold for all paths and cyclic graphs.
