Planarity of a unit graph: Part -II $ |Max(R)| = 2 $ case

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.

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.

International Electronic Journal of Algebra

Let $R$ be a commutative ring without identity. The zero-divisor graph of $R,$ denoted by $\Gamma(R)$ is a graph with vertex set $Z(R)\setminus \{0\}$ which is the set of all nonzero zero-divisor elements of $R,$ and two distinct vertices $x$ and $y$ are adjacent if and only if $xy=0.$ 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.

Mathematical Notes, 2015

Discrete Mathematics, Algorithms and Applications, 2016

Let [Formula: see text] be a commutative ring with identity. In this paper, we consider a simple graph associated with [Formula: see text] denoted by [Formula: see text], whose vertex set is the set of all nonzero proper ideals of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent whenever [Formula: see text] or [Formula: see text]. In this paper, we initiate the study of the graph [Formula: see text] and we investigate its properties. In particular, we show that [Formula: see text] is a connected graph with [Formula: see text] unless [Formula: see text] is isomorphic to a direct product of two fields. Moreover, we characterize all commutative rings [Formula: see text] with at least two maximal ideals for which [Formula: see text] are planar.

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.

Let R(+)N be the idealization of the ring R by the R-module N. In this paper, we investigate when Γ(R(+)N) is a Planar graph where R is an integral domain and we investigate when Γ(Z n (+)Z m) is a Planar graph.

Rocky Mountain Journal of Mathematics, 2018

We classify all finite groups with planar cyclic graphs. Also, we compute the genus and crosscap number of some families of groups (by knowing that of the cyclic graph of particular proper subgroups in some cases). 1. Introduction. Let G be a group. For each x ∈ G, the cyclizer of x is defined as Cyc G (x) = {y ∈ G | ⟨x, y⟩ is cyclic}. In addition, the cyclizer of G is defined by Cyc(G) = ∩ x∈G Cyc G (x). Cyclizers were introduced by Patrick and Wepsic in [15] and studied in [1, 2, 3, 9, 14, 15]. It is known that Cyc(G) is always cyclic and that Cyc(G) ⊆ Z(G). In particular, Cyc(G) G. The cyclic graph (respectively, weak cyclic graph) of a group G is the simple graph with vertex-set G\Cyc(G) (respectively, G\{1}) such that two distinct vertices x and y are adjacent if and only if ⟨x, y⟩ is cyclic. The cyclic graph and weak cyclic graph of G are denoted by Γ c (G) and Γ w c (G), respectively. From the explanation above, Γ c (G) (respectively, Γ w c (G)) is the null graph if G is cyclic (respectively, trivial). Thus, we will assume that G is non-cyclic (respectively, nontrivial) when working with Γ c (G) (respectively, Γ w c (G)). A graph is planar if it can be drawn in the plane in such a way that its edges intersect only at the end vertices. Recall that a subdivision of an edge {u, v} in a graph Γ is the replacement of the edge {u, v} in Γ with two new edges {u, w} and {w, v} in which w is a new vertex.

2013

This part of the Discussion forms a new bridge between the algebraic concept 'Ring', and 'Graph Theory'. We introduce 'Principal Ideal Graph', denoted by PIG(R), where R is a ring. The concept was introduced by Satyanarayana, Godloza and Nagaraju in the paper 'Some results on Principal ideal Graph of a ring', (published in African Journal of Mathematics and Computer Science Research, vol.4, 2011). We present some examples of PIG(ℤ ℤ ℤ ℤ n ) where ℤ ℤ ℤ ℤ n denotes the ring of integers modulo n for some values of n. We obtain fundamental important relations between rings and graphs with respect to the properties: simple ring, complete graph, Euler graph, etc.

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.

In this paper we characterize planar intersection graphs of ideals of a commutative ring with 1.

2016

We de ne a simple undirected graph PG1(R) with all the elements of a ring R as vertices, and two distinct vertices x, y are adjacent if and only if either x · y = 0 or y · x = 0 or x + y ∈ U(R), the set of all units of R. We have proved that PG1(Zn) is not Eulerian for any positive integer n. Also we discuss the Planarity and girth of PG1(R) and some cases which gives the degree of all vertices in PG1(R), over a ring Zn, for n ≤ 100.

Acta Mathematica Hungarica, 2012

Let R be a commutative ring with non-zero identity and G be a multiplicative subgroup of U (R), where U (R) is the multiplicative group of unit elements of R. Also, suppose that S is a non-empty subset of G such that S −1 = {s −1 | s ∈ S} S. Then we define Γ(R, G, S) to be the graph with vertex set R and two distinct elements x, y ∈ R are adjacent if and only if there exists s ∈ S such that x + sy ∈ G. This graph provides a generalization of the unit and unitary Cayley graphs. In fact, Γ(R, U (R), S) is the unit graph or the unitary Cayley graph, whenever S = {1} or S = {−1}, respectively. In this paper, we study the properties of the graph Γ(R, G, S) and extend some results in the unit and unitary Cayley graphs.

Let R be a commutative ring with unity. The co-maximal ideal graph of R, denoted by Γ(R), is a graph whose vertices are the proper ideals of R which are not contained in the Jacobson radical of R, and two vertices I1 and I2 are adjacent if and only if I1 + I2 = R. We classify all commutative rings whose co-maximal ideal graphs are planar. In 2012 the following question was posed: If Γ(R) is an infinite star graph, can R be isomorphic to the direct product of a field and a local ring? In this paper, we give an affirmative answer to this question. *

2019

Let R be a finite ring and r ∈ R. The r-noncommuting graph of R, denoted by ΓrR, is a simple undirected graph whose vertex set is R and two vertices x and y are adjacent if and only if [x, y] 6= r and −r. In this paper, we study several properties of ΓrR. We show that Γ r R is not a regular graph, a lollipop graph and complete bipartite graph. Further, we consider an induced subgraph of ΓrR (induced by the non-central elements of R) and obtained some characterizations of R.