The Coannihilator Graph of a Commutative Ring.

Let R be a commutative ring with nonzero identity. In this paper we intro- duce the coannihilator graph of R, which is a dual of the annihilator graph AG(R), denoted by AG'(R). AG'(R) is a graph with the vertex set W*(R), where W*(R) is the set of all nonzero and nonunit elements of R, and two distinct vertices x and y are adjacent if and only if x ∉ xyR or y ∉ xyR, where for z ∊ R, zR is the principal ideal generated by z. We study the interplay between the ring-theoretic properties of R and graph-theoretic properties of AG'(R). Also we completely determine all finite commutative rings R such that AG'(R) is planar, outerplanar or ring graph. Among other things, we prove that AG'(R) has a cut vertex if and only if R is isomorphic to Z∊2 × K, where K is a field. Also, we examine the domination number of AG'(R). [ABSTRACT FROM AUTHOR]