On the genus and crosscap two coannihilator graph of commutative rings

• Published in: Computational & applied mathematics Vol. 43; no. 6
• Main Authors: Nazim, Mohd, Mir, Shabir Ahmad, Rehman, Nadeem Ur
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.09.2024; Springer Nature B.V
• Subjects: Apexes; Applications of Mathematics; Commutativity; Computational Mathematics and Numerical Analysis; Graph theory; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Rings (mathematics); Vertex sets; 05C10; Cozero-divisor graph; Crosscap of a graph; Genus of a graph; 13M05; Coannihilator graph; 05C25; Zero-divisor graph
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-024-02872-7

Consider a commutative ring with unity denoted as R , and let W ( R ) represent the set of non-unit elements in R . The coannihilator graph of R , denoted as A G ′ ( R ) , is a graph defined on the vertex set W ( R ) ∗ . This graph captures the relationships among non-unit elements. Specifically, two distinct vertices, x and y , are connected in A G ′ ( R ) if and only if either x ∉ x y R or y ∉ x y R , where w R denotes the principal ideal generated by w ∈ R . In the context of this paper, the primary objective is to systematically classify finite rings 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 R for which the connihilator graph A G ′ ( R ) attains an outerplanarity index of two.