Rings whose total graphs have genus at most one

By H. Maimani, Cameron Wickham, Siamak Yassemi. 2010

83 citations recorded by the source corpus.

Let $R$ be a commutative ring with $\Z(R)$ its set of zero-divisors. In this paper, we study the total graph of $R$, denoted by $\T(\Gamma(R))$. It is the (undirected) graph with all elements of $R$ as vertices, and for distinct $x, y\in R$, the vertices $x$ and $y$ are adjacent if and only if $x + y\in\Z(R)$. We investigate properties of the total graph of $R$ and determine all isomorphism classes of finite commutative rings whose total graph has genus at most one (i.e., a planar or toroidal graph). In addition, it is shown that, given a positive integer $g$, there are only finitely many finite rings whose total graph has genus $g$.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/1ac30defb5724b87d475fbd9a220de2252a2ada8

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In Algebra and Number Theory

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Citations in the corpus

On the Total Graph and Its Complement of a Commutative Ring (2013, methodology): “[13], are used frequently in this article and hence given below.”
On the Genus of the Total Graph of a Commutative Ring (2013, methodology): “[13] determined the finite commutative rings R for which T R is a planar or toroidal graph.”