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- W4300374890 abstract "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 $x, yin R$, the vertices $x$ and $y$ are adjacent if and only if $x + yinZ(R)$. 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 $g$." @default.
- W4300374890 created "2022-10-03" @default.
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- W4300374890 date "2010-01-29" @default.
- W4300374890 modified "2023-09-27" @default.
- W4300374890 title "Rings whose total graphs have genus at most one" @default.
- W4300374890 doi "https://doi.org/10.48550/arxiv.1001.5338" @default.
- W4300374890 hasPublicationYear "2010" @default.
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