The Generalized Total Graph of a Commutative Ring (original) (raw)

On the Associated Graphs to a Commutative Ring

Journal of Algebra and Its Applications, 2012

Let R be a commutative ring with nonzero identity. For an arbitrary multiplicatively closed subset S of R, we associate a simple graph denoted by ΓS(R) with all elements of R as vertices, and two distinct vertices x, y ∈ R are adjacent if and only if x+y ∈ S. Two well-known graphs of this type are the total graph and the unit graph. In this paper, we study some basic properties of ΓS(R). Moreover, we will improve and generalize some results for the total and the unit graphs.

Total Graphs Associated to a Commutative Ring

2016

LetR be a commutative ring with nonzero unity. Let Z(R) be the set of all zerodivisors ofR. The total graph of R, denoted byT (Γ(R)), is the simple graph with vertex set R and two distinct verticesx andy are adjacent if their sumx + y ∈ Z(R). Several authors presented various generalizations for T (Γ(R)). This article surveys research conducted on T (Γ(R)) and its generalizations. A historical review of literature is given. Further p roperties ofT (Γ(R)) are also studied. Many open problems are presented for further rese arch.

The total graph of a commutative ring," Journal of Algebra 320 (2008) 2706–2719

2008

Let R be a commutative ring with Nil(R) its ideal of nilpotent elements, Z(R) its set of zero-divisors, and Reg(R) its set of regular elements. In this paper, we introduce and investigate 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 ∈ R, the vertices x and y are adjacent if and only if x + y ∈ Z(R). We also study the three (induced) subgraphs Nil(Γ (R)), Z(Γ (R)), and Reg(Γ (R)) of T (Γ (R)), with vertices Nil(R), Z(R), and Reg(R), respectively.

A Generalization of Total Graphs of Modules over Commutative Rings under Multiplicatively Closed Subsets

Journal of Mathematical Extension, 2017

Let R be a commutative ring and M be an R -module with a proper submodule N . A generalization of total graphs, denoted by T (Γ N H ( M )), is introduced and investigated. It is the (undirected) graph with all elements of M as vertices and for distinct x; y 2 M , the vertices x; y are adjacent if and only if x + y 2 M H ( N ) where M H ( N ) = f m 2 M : rm 2 N for some r 2 H g and H is a multiplicatively closed subset of R . In this paper, in addition to studying some algebraic properties of M H ( N ), we investigate some graph theoretic properties of two essential subgraphs of T (Γ N H ( M )).

On the Total Graph of a Ring and Its Related Graphs: A Survey, In Commutative Algebra: Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, DOI 10.1007/978-1-4939-0925-4__3, edited by M. Fontana et al. (eds.),, 39-54. New York: Springer, 2014.

2014

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. .R// with vertices all elements of R, and two distinct vertices x and y are adjacent if and only if x C y 2 Z.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.