Cohen-Macauleyness of the Zero-Divisor Graph of a Boolean Poset

In this paper, we prove that the zero-divisor graph $\mathrm{\Gamma }(P)$ of a Boolean poset $P$ is both well-covered and Cohen–Macaulay. Furthermore, for a poset $\mathbf{P}={\prod }_{i=1}^n{P}_i$ $(n\ge 3)$, where each $P_i$ is a finite bounded poset satisfying $Z(P_i)=\{0\}$ for all $i$, and $\le \mathrm{\mid }{P}_1\mathrm{\mid }\le \mathrm{\mid }{P}_2\mathrm{\mid }\le \cdots \le \mathrm{\mid }{P}_n\mathrm{\mid },$ we show that the zero-divisor graph $\mathrm{\Gamma }(\mathbf{P})$ is Cohen–Macaulay if and only if $\mathbf{P}$ is a Boolean lattice.