Let R be a Noetherian commutative ring. The regular digraph of ideals of R introduced in [10] which denoted by −−→ Γreg(R). The vertex set of −−→ Γreg(R) is set of nontrivial ideals of R and I −→ J is an arc in Γ(R) if and only if I ⊈ Z(J). In this paper we will introduce a dual of −−→ Γreg(R) and denote it by Γ∗(R). The vertex set of Γ∗(R) is set of nontrivial ideals of R and I −→ J is an arc in Γ∗(R) if and only if there exists an element x in I such that the R−module homomorphism fx : J −→ J is an epimorphism by the rule fx(y) = yx. We investigate the interplay between Γ∗(R) and ring-theoretic properties of R and graph-theoretic properties of Γ∗(R). Also we study the relation between two graphs Γ(R) and Γ∗(R). In particular we will show that the value of !(Γreg(R)) in [10] is not correct, and we will compute the correct value of w(Γreg(R)).

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