Vertex-decomposable, Cohen--Macaulay and well-covered zero-divisor graphs

Let R be a finite commutative ring with 1 ≠ 0. The zero-divisor graph of R is the graph obtained by letting all the nonzero zero-divisors of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if xy = 0. In this paper, vertex-decomposability, Cohen–Macaulayness and well-coveredness of zero-divisor graphs are characterized.