On the genus and crosscap two coannihilator graph of commutative rings

Consider a commutative ring with unity denoted as \(\mathscr {R}\), and let \(W(\mathscr {R})\) represent the set of non-unit elements in \(\mathscr {R}\). The coannihilator graph of \(\mathscr {R}\), denoted as \(AG'(\mathscr {R})\), is a graph defined on the vertex set \(W(\mathscr {R})^*\). This graph captures the relationships among non-unit elements. Specifically, two distinct vertices, x and y, are connected in \(AG'(\mathscr {R})\) if and only if either \(x \notin xy\mathscr {R}\) or \(y \notin xy\mathscr {R}\), where \(w\mathscr {R}\) denotes the principal ideal generated by \(w \in \mathscr {R}\). In the context of this paper, the primary objective is to systematically classify finite rings \(\mathscr {R}\) based on distinct characteristics of their coannihilator graph. The focus is particularly on cases where the coannihilator graph exhibits a genus or crosscap of two. Additionally, the research endeavors to provide a comprehensive characterization of finite rings \(\mathscr {R}\) for which the connihilator graph \(AG'(\mathscr {R})\) attains an outerplanarity index of two.