Some Properties of Von Neumann Regular Graphs of Rings

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).

Asian Research Journal of Mathematics

For a nontrivial connected graph G with no isolated vertex, a nonempty subset D \(\subseteq\) V (G) is a rings dominating set if D is a dominating set and for each vertex \(\upsilon\) \(\in\) V \ D is adjacent to at least two vertices in V \ D. Thus, the dominating set D of V (G) is a rings dominating set if for all \(\upsilon\) \(\in\) V \ D, \(\mid\)N(\(\upsilon\)) \(\cap\) (V \ D)\(\mid\) \(\ge\) 2. Moreover, D is called a minimum rings dominating set if D is a rings dominating set of smallest size in a given graph. The cardinality of minimum rings dominating set of G is the rings domination number of G, denoted by \(\gamma\)ri(G). Here, we determine how the minimum rings dominating set is constructed in the total graph of some graph families with the inclusion of generated conditions for this type of domination and provide their respective rings domination number.

Communications in Algebra, 2013

Algebra Colloquium, 2012

Let R be a commutative ring with nonzero identity. In this paper, we study the von Neumann regular elements of R. We also study the idempotent elements, π-regular elements, the von Neumann local elements, and the clean elements of R. Finally, we investigate the subgraphs of the zero-divisor graph Γ(R) of R induced by the above elements.

2012

Let R be a commutative ring with nonzero identity. In this paper, we study the von Neumann regular elements of R. We also study the idempotent elements, π-regular elements, the von Neumann local elements, and the clean elements of R. Finally, we investigate the subgraphs of the zero-divisor graph Γ(R) of R induced by the above elements.

Indian Journal of Pure and Applied Mathematics

The graph Γ is the simple undirected graph with vertex set R in which two distinct vertices x and y are adjacent if and only if there exists a unit element u in U (R) such that x + uy is a unit in R. Also, Γ denotes the complement of Γ. In this paper, we find the domination number γ of Γ as well as Γ and characterize all γ-sets in Γ and Γ. Also, we obtain the bondage number of Γ. Further, we obtain the values of some domination parameters like independent, strong and weak domination numbers of Γ.

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.

Discrete Mathematics, Algorithms and Applications, 2016

The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let R be a ring. We denote the collection of all ideals of R by I(R) and I(R)\{(0)} by I(R) *. Alilou et al. [A. Alilou, J. Amjadi and S.M. Sheikholeslami, A new graph associated to a commutative ring, Discrete Math. Algorithm. Appl. 8 (2016) Article ID: 1650029 (13 pages)] introduced and investigated a new graph associated to R, denoted by Ω * R which is an undirected graph whose vertex set is I(R) * \{R} and distinct vertices I, J are joined by an edge in this graph if and only if either (Ann R I)J = (0) or (Ann R J)I = (0). Several interesting theorems were proved on Ω * R in the aforementioned paper and they illustrate the interplay between the graph-theoretic properties of Ω * R and the ring-theoretic properties of R. The aim of this article is to investigate some properties of (Ω * R) c , the complement of the new graph Ω * R associated to 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.

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)).