A note on zero-divisor graph of amalgamated duplication of a ring along an ideal

摘要

Let R be a commutative ring and I be a non-zero ideal of R . Let R ? I be the subring of R × R consisting of the elements ( r , r + i ) for r ∈ R and i ∈ I . In this paper we characterize all isomorphism classes of finite commutative rings R with identity and ideal I such that Γ ( R ? I ) is planar. We determine the number of vertices of Γ ( R ? I ) , a necessary and sufficient condition for the graph Γ ( R ? I ) to be outerplanar and the domination number of Γ ( R ? I ) .