Let $n\ge 2$ be an integer. The graph $G(n)$ is obtained by letting all the elements of $\{0,\dots ,n-1\}$ to be the vertices and defining distinct vertices x and y to be adjacent if and only if $\gcd (x+y,n)=1$. In this paper, well-coveredness, Cohen–Macaulayness, vertex-decomposability and Gorensteinness of these graphs and their complements are characterized. These characterizations provide large classes of Cohen–Macaulay and non Cohen–Macaulay graphs.

中文翻译：

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一类图的 Cohen-Macaulayness 与它们的补图类

让 $n\ge 2$是一个整数。图$G(n)$ 是通过让所有元素获得 $\{0,\dots ,n-1\}$成为顶点并定义不同的顶点x和y相邻当且仅当$\mathrm{GCD}(X+是,n)=1$. 在本文中，这些图及其补集的良好覆盖性、Cohen-Macaulayness、顶点可分解性和Gorensteinness 被表征。这些特征提供了大量的 Cohen-Macaulay 图和非 Cohen-Macaulay 图。