Let \((P, \le )\) be a partially ordered set with minimum element 0 and for every \(x, y \in P\) set \(L(x,y) = \{z \in P \mid z \le x \text { and } z \le y\}\) . An element \(x \in P\) is called a zero-divisor of P if \(L(x,y) = \{ 0 \}\) for some \(0 \ne y \in P\) . The zero-divisor graph \(\Gamma (P)\) of P is obtained by letting all nonzero zero-divisors of P to be the vertices and defining distinct vertices x and y to be adjacent if and only if \(L(x,y) = \{ 0 \}\) . In this paper, we give a comprehensive characterization of well-coveredness, very well-coveredness, Cohen–Macaulayness and Gorensteinness for the zero-divisor graphs of divisor posets.
