In this paper, we introduce some new definitions such as the U* L* condition to describe the zero-divisor graph G=Γ(P) of a poset P, and give a new and quick proof to a main result in [2, 4]. By deleting a typical vertex with least degree, we provide an algorithm for finding a maximum clique of a finite graph G. We study some properties of the zero-divisor graphs of posets concerning diameters and girths. We also provide stratified presentations of posets.