Abstract
In this paper, we consider the problem of classifying commutative rings according to the crosscap number of it’s associating graphs. In particular, we enumerate all isomorphism classes of commutative rings with identity whose zero-divisor graph and total graph have crosscap two respectively. This answers a case of a conjecture posed by Chiang-Hsieh in (2008) and Khashyarmanesh et al. in (2013).
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