图边理想的Cohen-Macaulay性质及代数不变量

组合交换代数中的Cohen-Macaulay性质（简称CM性质）以其丰富与深刻的代数内涵而广受关注；而顶点可分解性质因其特有的归纳属性，成为研究shellability以及（序列）CM性质的重要途径。本项目拟结合代数方法与图论结构分析，从多个维度来探讨顶点可分解的组合属性与图边理想CM性质之间更多的深刻联系，充分利用图结构的直观性、组合技巧性和代数的深刻性，挖掘出更多的CM图类。通过本项目的研究，我们将解决与顶点可分解性质相关的组合与代数问题，进一步完善CM图的分类工作。

Cohen-Macaulay property (referred to as CM property) in combinatorial commutative algebra is widely concerned by its rich and profound algebraic connotation. Meanwhile the vertex decomposablility is an important way to study shellability and (sequentially) CM property due to its unique inductive properties. This project intends to combine algebraic methods and the structure analysis of graph to explore more and deeper connections between combinatorial attributes such as vertex decomposability and CM property of edge ideals from multiple dimensions, and makes full use of the intuitiveness structure of graphs, the skills of combination and the profoundness of algebra, to dig out more CM graph classes. Through this project, we hope to solve the combinatorial and algebraic problems related to vertex decomposablility. Furthermore, we hope to improve the classification of CM graphs.