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<p>Characterizations of the classes of Cohen–Macaulay graphs are important because when we characterize one of these classes, then we can decide whether a ring of the form 𝕂[<i>x</i><sub>1</sub>,…,<i>x</i><sub><i>n</i></sub>]/<i>I</i> is Cohen–Macaulay or not, where <i>I</i> is a square-free monomial ideal. For a given commutative ring <i>R</i>, the total graph of <i>R</i> is a simple graph with <i>R</i> as the vertex set and two distinct vertices <i>x</i> and <i>y</i> are adjacent if <i>x</i> + <i>y</i> is a zero-divisor of <i>R.</i> We find two classes of the total graphs that are not Cohen–Macaulay classes.</p>

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Classes of Graphs That Are Not Cohen–Macaulay Classes

  • 期刊论文

  • 发表日期：

    2025年7月1日

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  • T. Asir,
  • T. Ashitha

摘要

Characterizations of the classes of Cohen–Macaulay graphs are important because when we characterize one of these classes, then we can decide whether a ring of the form 𝕂[x1,…,xn]/I is Cohen–Macaulay or not, where I is a square-free monomial ideal. For a given commutative ring R, the total graph of R is a simple graph with R as the vertex set and two distinct vertices x and y are adjacent if x + y is a zero-divisor of R. We find two classes of the total graphs that are not Cohen–Macaulay classes.