The zero-divisor graph of a reduced ring

In this paper the zero-divisor graph Γ(R)Γ(R) of a commutative reduced ring RR is studied. We associate the ring properties of RR, the graph properties of Γ(R)Γ(R) and the topological properties of Spec(R). Cycles in Γ(R)Γ(R) are investigated and an algebraic and a topological characterization is given for the graph Γ(R)Γ(R) to be triangulated or hypertriangulated. We show that the clique number of Γ(R)Γ(R), the cellularity of Spec(R) and the Goldie dimension of RR coincide. We prove that when RR has the annihilator condition and 2∉Z(R); Γ(R)Γ(R) is complemented if and only if Min(R) is compact. In a semiprimitive Gelfand ring, it turns out that the dominating number of Γ(R)Γ(R) is between the density and the weight of Spec(R). We show that Γ(R)Γ(R) is not triangulated and the set of centers of Γ(R)Γ(R) is a dominating set if and only if the set of isolated points of Spec(R) is dense in Spec(R).