On the Associated Graphs to a Commutative Ring

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).

Journal of Algebra and Its Applications, 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).

Communications in Algebra, 2013

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.

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.

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)).

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.

arXiv preprint arXiv:1108.2863, 2011

Let R be a ring (not necessary commutative) with non-zero identity. The unit graph of R, denoted by G(R), is a graph with elements of R as its vertices and two distinct vertices a and b are adjacent if and only if a + b is a unit element of R. It was proved that if R is a commutative ring and m is a maximal ideal of R such that |R/m| = 2, then G(R) is a complete bipartite graph if and only if (R, m) is a local ring. In this paper we generalize this result by showing that if R is a ring (not necessary commutative), then G(R) is a complete r-partite graph if and only if (R, m) is a local ring and r = |R/m| = 2 n , for some n ∈ N or R is a finite field. Among other results we show that if R is a left Artinian ring, 2 ∈ U (R) and the clique number of G(R) is finite, then R is a finite ring.

Communications in Algebra, 2010

Let R be a ring with nonzero identity. The unit graph of R, denoted by G R , has its set of vertices equal to the set of all elements of R; distinct vertices x and y are adjacent if and only if x + y is a unit of R. In this article, the basic properties of G R are investigated and some characterization results regarding connectedness, chromatic index, diameter, girth, and planarity of G R are given. (These terms are defined in Definitions and Remarks 4

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.