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Paper   IPM / M / 11463
School of Mathematics
  Title:   Rings whose total graphs have genus at most one
1.  H. R. Maimani
2.  S. Yassemi (Joint with C. Wickham)
  Status:   Published
  Journal: Rocky Mountain J. Math.
  Year:  2012
  Pages:   DOI: 10.1216/RMJ-2012-42-5-1
  Supported by:  IPM
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(Γ(R)). It is the (undirected) graph with all elements of R as vertices, and for distinct x, yR, the vertices x and y are adjacent if and only if x + yZ(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.

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