Rings whose total graphs have genus at most one

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(Γ(R)). It is the (undirected) graph with all elements of R as vertices and, for distinct x, y ε R, the vertices x and y are adjacent if and only if x + y ε 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.

Department(s)

Mathematics

Document Type

Article

DOI

https://doi.org/10.1216/RMJ-2012-42-5-1551

Keywords

Genus, Planar graph, Toroidal graph, Total graph

Publication Date

12-1-2012

Journal Title

Rocky Mountain Journal of Mathematics

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