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Let  S be a commutative ring with unity, and a set of nonunit elements is denoted by  W S . The coannihilator graph of  S , denoted by  A G ′ S , is an undirected graph with vertex set  W S ∗ (set of all nonzero nonunit elements of  S ), and  α ∼ β is an edge of  A G ′ S ⇔ α ∉ α β S or  β ∉ α β S , where  δ S denotes the principal ideal generated by  δ ∈ S . In this study, we first classify finite ring  S , for which  A G ′ S is isomorphic to some well-known graph. Then, we characterized the finite ring  S , for which  A G ′ S is toroidal or projective.

Classification of Rings with Toroidal and Projective Coannihilator Graph