Planarity of a unit graph part -III $ |Max(R)| \geq 3 $ case (original) (raw)
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Planarity of a unit graph: Part -II $ |Max(R)| = 2 $ case
2020
The rings considered in this article are commutative with identity 1 6= 0. Recall that the unit graph of a ring R is a simple undirected graph whose vertex set is the set of all elements of the ring R and two distinct vertices x,y are adjacent in this graph if and only if x+ y ∈U(R) where U(R) is the set of all unit elements of ring R. We denote this graph by UG(R). In this article we classified rings R with |Max(R)|= 2 such that UG(R) is planar.
Planarity of a unit graph: Part-I local case
2020
The rings considered in this article are commutative with identity 1 6= 0. Recall that the unit graph of a ring R is a simple undirected graph whose vertex set is the set of all elements of the ring R and two distinct vertices x,y are adjacent in this graph if and only if x+ y ∈U(R) where U(R) is the set of unit elements of ring R. We denote this graph by UG(R). In this article we classified local ring R such that UG(R) is planar.
On the Planarity of Extended Zero-Divisor Graphs of Local Rings
Mathematical Problems in Engineering, 2022
Let R be a commutative ring with unity 1 ≠ 0 . Recently Bennis et al. defined the concept of extended zero-divisor graph Γ ¯ R by considering the vertex set V Γ ¯ R = Z ∗ R and any two vertices x and y are adjacent if there exist positive integers m and n , such that x m y n = 0 with x m ≠ 0 and y n ≠ 0 . The main objective of this article is to check the planar property of extended zero-divisor graphs. Also, a complete list of local rings up to order 27 with planar extended zero-divisor graphs has been collected.
When a zero-divisor graph is planar or a complete r-partite graph
Journal of Algebra, 2003
Let Γ(R) be the zero-divisor graph of a commutative ring R. An interesting question was proposed by Anderson, Frazier, Lauve, and Livingston: For which finite commutative rings R is Γ(R) planar? We give an answer to this question. More precisely, we prove that if R is a local ring with at least 33 elements, and Γ(R)≠∅, then Γ(R) is not planar.
Unit Graphs Associated with Rings
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
2018
The rings considered in this article are nonzero commutative with identity which are not fields. Let \(R\) be a ring. We denote the collection of all proper ideals of \(R\) by \(\mathbb{I}(R)\) and the collection \(\mathbb{I}(R)\setminus \{(0)\}\) by \(\mathbb{I}(R)^{*}\). Recall that the intersection graph of ideals of \(R\), denoted by \(G(R)\), is an undirected graph whose vertex set is \(\mathbb{I}(R)^{*}\) and distinct vertices \(I, J\) are adjacent if and only if \(I\cap J\neq (0)\). In this article, we consider a subgraph of \(G(R)\), denoted by \(H(R)\), whose vertex set is \(\mathbb{I}(R)^{*}\) and distinct vertices \(I, J\) are adjacent in \(H(R)\) if and only if \(IJ\neq (0)\). The purpose of this article is to characterize rings \(R\) with at least two maximal ideals such that \(H(R)\) is planar.
On the Unit Graph of a Noncommutative Ring
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.
Planar index and outerplanar index of zero-divisor graphs of commutative rings without identity
International Electronic Journal of Algebra
Let RRR be a commutative ring without identity. The zero-divisor graph of R,R,R, denoted by Gamma(R)\Gamma(R)Gamma(R) is a graph with vertex set Z(R)setminus0Z(R)\setminus \{0\}Z(R)setminus0 which is the set of all nonzero zero-divisor elements of R,R,R, and two distinct vertices xxx and yyy are adjacent if and only if xy=0.xy=0.xy=0. In this paper, we characterize the rings whose zero-divisor graphs are ring graphs and outerplanar graphs. Further, we establish the planar index, ring index and outerplanar index of the zero-divisor graphs of finite commutative rings without identity.
Classification of rings with unit graphs having domination number less than four
Rendiconti del Seminario Matematico della Università di Padova, 2015
Let R be a finite commutative ring with nonzero identity. The unit graph of R is the graph obtained by setting all the elements of R to be the vertices and defining distinct vertices x and y to be adjacent if and only if x + y is a unit element of R. In this paper, a classification of finite commutative rings with nonzero identity in which their unit graphs have domination number less than four is given.