On the domination and signed domination numbers of zero-divisor graph

Abstract

Let R be a commutative ring (with 1) and let Z(R) be its set of zero-divisors. The zero-divisor graph Γ(R) has vertex set Z*(R) = Z(R) \ [0] and for distinct x,y ε Z*(R), the vertices x and y are adjacent if and only if xy = 0. In this paper, we consider the domination number and signed domination number on zero-divisor graph Γ(R) of commutative ring R such that for every 0 ≠ x ε Z*(R), x2 ≠ 0. We characterize Γ(R) whose γ(Γ(R)) + γ(Γ(R)) ε [n + 1, n, n-1], where |Z*(R)|=n.