Classes of graphs that are not Cohen–Macaulay classes - Asir, T.; Ashitha, T. (Ukrains’kyi Matematychnyi Zhurnal, 2025)
UDC 512.5Characterizations 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 $\mathbb{K}[x_1,\ldots,x_n]/I$ 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.