Abstract
Let S be a commutative ring with unity, and a set of nonunit elements is denoted by W(G). The coannihilator graph of S, denoted by AG '(G), is an undirected graph with vertex set W(S)* (set of all nonzero nonunit elements of S), and alpha similar to beta is an edge of AG '(S)double left right arrow alpha is not an element of alpha beta S or beta is not an element of alpha beta S, where delta S denotes the principal ideal generated by delta is an element of S. In this study, we first classify finite ring S, for which AG '(S)double left right arrow alpha is not an element of alpha beta Sopen is isomorphic to some well-known graph. Then, we characterized the finite ring S, for which AG '(S) is toroidal or projective.