The total graph and regular graph of a commutative ring

Hacettepe Journal of Mathematics and Statistics

Let R be a commutative ring with unity. The total graph of R, T (Γ(R)), is the simple graph with vertex set R and two distinct vertices are adjacent if their sum is a zero-divisor in R. Let Reg(Γ(R)) and Z(Γ(R)) be the subgraphs of T (Γ(R)) induced by the set of all regular elements and the set of zero-divisors in R, respectively. We determine when each of the graphs T (Γ(R)), Reg(Γ(R)), and Z(Γ(R)) is locally connected, and when it is locally homogeneous. When each of Reg(Γ(R)) and Z(Γ(R)) is regular and when it is Eulerian.

2012

Let R be a commutative ring with nonzero identity, and let Z(R) be its set of zerodivisors. The total graph of R is the (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 + y ∈ Z(R). In this paper, we study the two (induced) subgraphs Z 0 (Γ(R)) and T 0 (Γ(R)) of T (Γ(R)), with vertices Z(R)\{0} and R\{0}, respectively. We determine when Z 0 (Γ(R)) and T 0 (Γ(R)) are connected and compute their diameter and girth. We also investigate zerodivisor paths and regular paths in T 0 (Γ(R)).

Communications in Algebra, 2013

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.

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.

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.

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.

Journal of the Korean Mathematical Society, 2012

Let R be a commutative ring and I its proper ideal, let S(I) be the set of all elements of R that are not prime to I. Here we introduce and study the total graph of a commutative ring R with respect to proper ideal I, denoted by T (Γ I (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 ∈ S(I). The total graph of a commutative ring, that denoted by T (Γ(R)), is the graph where the vertices are all elements of R and where there is an undirected edge between two distinct vertices x and y if and only if x + y ∈ Z(R) which is due to Anderson and Badawi [2]. In the case I = {0}, T (Γ I (R)) = T (Γ(R)); this is an important result on the definition.

2013

Let R be a commutative ring with nonzero identity and H be a nonempty proper subset of R such that R\H is a saturated multiplicatively closed subset of R. The generalized total graph of R is the (simple) graph GT H (R) with all elements of R as the vertices, and two distinct vertices x and y are adjacent if and only if x + y ∈ H. In this paper, we investigate the structure of GT H (R).

2022

We give a decomposition of total graphs on some finite commutative rings R = Zm, where the set of zero-divisors of R is not an ideal. In particular, we study the total graph T(􀀀(Z2npm))where p is a prime and m and n are positive integers and investigate some graph theoretical properties with some of its fundamental subgraphs.