The Coannihilator Graph of a Commutative Ring

Let R be a commutative ring with nonzero identity. In this paper we introduce 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 is not an element of 2 xyR or y is not an element of 2 xyR, where for z is an element of 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 fi nite 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) x K, where K is a field. Also, we examine the domination number of AG' (R).