IJAM, Volume 26, No. 1 (2013)

Let R be a commutative ring and let Γ(Z_n) be the zero divisor graph of R. The zero-divisor graph of a ring is the graph(simple) whose vertex set is the set of non-zero zero-divisors, and an edge is drawn between two distinct vertices if their product is zero. For a graph Γ(Z_n), a set $S⊆V(Γ(Z_n)) is a weak dominating set if every vertex v in V(Γ(Z_n))-S has a neighbour u in S such that the degree of u is not greater than the degree of v. The minimum cardinality of a weak dominating set of Γ(Z_n) is the weak domination number, γ_w(Γ(Z_n)).

In this paper, we present some bounds on Γ(Z_n) and give exact values for γ_w(Γ(Z_2p)), γ_w(Γ(Z_p²)), γ_w(Γ(Z_pq)), γ_w(Γ(Z₂^np})), γ_w(Γ(Z₃n^p)) and γ_w(Γ(Z_pqr)).