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

Chapter

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 

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsAmerican University of SharjahSharjahUnited Arab Emirates

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