Some Properties of Von Neumann Regular Graphs of Rings

Journal of Combinatorial Mathematics and Combinatorial Computing, 2014

Let R be a commutative ring and Z(R) be its set of all zerodivisors. The total graph of R, denoted by TΓ(R), is the undirected graph with vertex set R, and two distinct vertices x and y are adjacent if and only if x + y ∈ Z(R). In this paper, we obtain a lower bound as well as an upper bound for domination number of TΓ(R). Further we proved that the upper bound for the domination number of TΓ(R) is attained in the case an Artin ring R. Having proved this, we have identified certain classes of rings corresponding to which the domination number of the total graph equals the upper bound. In view of these assertions, we conjecture that the domination number equals to this upper bound. Certain other domination parameters are also obtained for TΓ(R) under the assumption that the conjecture is true.

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.

Journal of Mathematics

One of the most important branches of mathematics is algebraic graph theory, which solves graph problems with algebraic methods. In graph theory, several algebraic properties of a ring can be represented. In this paper, we define an innovative graph on rings, explore its characteristics, and examine how it relates to other notions in the field. Let S be a ring; the quasi-regular graph of S is a graph with a vertex set of S − 0 and any two different vertices w and z are adjacent if 1 − w z is a unit in S . We study this graph by providing different examples and proving some crucial characteristics. This study provides important results and paves the way for a lot of different inquiries and studies utilizing this novel approach.

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 Pure and Applied Algebra, 2009

Communicated by J. Walker MSC: 05C40 05C45 16P10 16P40

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.