Geodesic
On the Genus of the Co-Annihilating Graph of Commutative Rings
Discussiones Mathematicae. General Algebra and Applications, Tome 39 (2019) no. 2, pp. 203-220

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Let R be a commutative ring with identity and 𝔘_R be the set of all nonzero non-units of R. The co-annihilating graph of R, denoted by 𝒞𝒜_R, is a graph with vertex set 𝔘_R and two vertices x and y are adjacent whenever ann(x) ∩ ann(y) = (0). In this paper, we characterize all commutative Artinian non-local rings R for which the 𝒞𝒜_R has genus one and two. Also we characterize all commutative Artinian non-local rings R for which 𝒞𝒜_R has crosscap one. Finally, we characterize all finite commutative non-local rings for which g(Г_2(R)) = g(𝒞𝒜_R) = 0 or 1.
Keywords: co-annihilating graph, planar graph, genus, crosscap 
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Selvakumar, K.; Karthik, S. On the Genus of the Co-Annihilating Graph of Commutative Rings. Discussiones Mathematicae. General Algebra and Applications, Tome 39 (2019) no. 2, pp. 203-220. http://geodesic.mathdoc.fr/item/DMGAA_2019_39_2_a3/