Cohen–macauleyness of the Zero-Divisor Graph of a Boolean Poset

We prove that the zero-divisor graph Γ (P) of a Boolean poset P is both well-covered and Cohen–Macaulay. Furthermore, for a poset P = ∏ ᵢ₌₁ⁿ Pᵢ \ (n ≥ 3) , where each Pᵢ is a finite bounded poset satisfying Z(Pᵢ) = \0\ for all i and 2 ≤ |P₁| ≤ |P₂| ≤ ⋯ ≤ |Pₙ|, we show that the zero-divisor graph Γ (P) is Cohen–Macaulay if and only if P is a Boolean lattice.