Chapter

Commutative Algebra

pp 39-54

Date:

On the Total Graph of a Ring and Its Related Graphs: A Survey

  • Ayman BadawiAffiliated withDepartment of Mathematics and Statistics, American University of Sharjah Email author 

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Abstract

Let R be a (commutative) ring with nonzero identity and Z(R) be the set of all zero divisors of R. The total graph of R is the simple undirected graph T(Γ(R)) with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x + yZ(R). This type of graphs has been studied by many authors. In this paper, we state many of the main results on the total graph of a ring and its related graphs.

Keywords

Total graph Zero divisors Diameter Girth Connected graph Genus Generalized total graph Dominating set Clique Chromatic number

MSC(2010) classification

13A15 13B99 05C99