Classes of graphs that are not Cohen–Macaulay classes

UDC 512.5 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 $\mathbb{K}[x_1,\ldots,x_n]/I$ 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.