Ideal-based Zero-divisor Graph Research Articles

Harary and hyper-Wiener indices of some graph operations

In this paper, we obtain the Harary index and the hyper-Wiener index of the H-generalized join of graphs and the generalized corona product of graphs. As a consequence, we deduce some of the results in (Das et al. in J. Inequal. Appl. 2013:339, 2013) and (Khalifeh et al. in Comput. Math. Appl. 56:1402–1407, 2008). Moreover, we calculate the Harary index and the hyper-Wiener index of the ideal-based zero-divisor graph of a ring.

Padmakar–Ivan index of some families of graphs

In this paper, we obtain the vertex Padmakar–Ivan index of [Formula: see text]-generalized join of graphs. As a consequence, we find the vertex Padmakar–Ivan index of the lexicographic product of graphs, the ideal-based zero-divisor graph and the zero-divisor graph of rings. In addition, we obtain the vertex Padmakar–Ivan index of the ideal-based zero-divisor graph and the zero-divisor graph of the ring of integers modulo [Formula: see text].

Wiener index of an ideal-based zero-divisor graph of commutative ring with unity

The Wiener index of a connected graph G is W ( G ) = ∑ { u , v } ⊆ V ( G ) d G ( u , v ) . In this paper, we obtain the Wiener index of H-generalized join of graphs G 1 , G 2 , … , G k . As a consequence, we obtain some earlier known results in [Alaeiyan et al. in Aust. J. Basic Appl. Sci. (2011) 5(12): 145–152; Yeh et al. in Discrete Math. (1994) 135: 359–365] and we also obtain the Wiener index of the generalized corona product of graphs. We further show that the ideal-based zero-divisor graph Γ I ( R ) is a H-generalized join of complete graphs and totally disconnected graphs. As a result, we find the Wiener index of the ideal-based zero-divisor graph Γ I ( R ) and we deduce some of the main results in [Selvakumar et al. in Discrete Appl. Math. (2022) 311: 72–84]. Moreover, we show that W ( Γ I ( Z n ) ) is a quadratic polynomial in n, where Z n is the ring of integers modulo n and we calculate the exact value of the Wiener index of Γ Nil ( R ) ( R ) , where Nil(R) is nilradical of R. Furthermore, we give a Python program for computing the Wiener index of Γ I ( Z n ) if I is an ideal of Z n generated by pr , where pr is a proper divisor of n, p is a prime number and r is a positive integer with r ≥ 2 .

On the ideal-based zero-divisor graph of commutative rings

Let [Formula: see text] be a finite commutative ring with identity, [Formula: see text] be an ideal of [Formula: see text] and [Formula: see text] denotes the Jacobson radical of [Formula: see text]. The ideal-based zero-divisor graph [Formula: see text] of [Formula: see text] is a graph with vertex set [Formula: see text] for some [Formula: see text] in which distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we determine the diameter, girth of [Formula: see text]. Specifically, we classify all finite commutative nonlocal rings for which [Formula: see text] is perfect. Furthermore, we discuss about the planarity, outerplanarity, genus and crosscap of [Formula: see text] and characterize all of them.

An ideal-based cozero-divisor graph of a commutative ring

Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this article, we introduce the cozero-divisor graph $\acute{\Gamma}_I(R)$ of $R$ and explore some of its basic properties. This graph can be regarded as a dual notion of an ideal-based zero-divisor graph.

On the ideal-based zero-divisor graphs

Let R be a commutative ring. In this paper, we study the annihilator ideal-based zero-divisor graph by replacing the ideal I of R with the ideal AnnR(M) for an R-module M. Also, we investigate a certain subgraph of the annihilator ideal-based zero-divisor graph and obtain some related results.

WHEN IDEAL-BASED ZERO-DIVISOR GRAPHS ARE COMPLEMENTED OR UNIQUELY COMPLEMENTED

Let R be a commutative ring with nonzero identity and I a proper ideal of R. The ideal-based zero-divisor graph of R with respect to the ideal I, denoted by ΓI (R), is the graph on vertices {x ∈ R \ I | xy ∈ I for some y ∈ R\I}, where distinct vertices x and y are adjacent if and only if xy ∈ I. In this paper, we give a complete classification of when an ideal-based zero-divisor graph of a commutative ring is complemented or uniquely complemented based on the total quotient ring of R/I.

A GENERAL THEORY OF ZERO-DIVISOR GRAPHS OVER A COMMUTATIVE RING

Let R be a commutative ring with 1 6= 0, I a proper ideal of R, and ∼ a multiplicative congruence relation on R. Let R/∼ = { [x]∼ | x ∈ R } be the commutative monoid of ∼-congruence classes under the induced multiplication [x]∼[y]∼ = [xy]∼, and let Z(R/∼) be the set of zero-divisors of R/∼. The ∼-zero-divisor graph of R is the (simple) graph Γ∼(R) with vertices Z(R/∼) \{[0]∼} and with distinct vertices [x]∼ and [y]∼ adjacent if and only if [x]∼[y]∼ = [0]∼. Special cases include the usual zero-divisor graphs Γ(R) and Γ(R/I), the ideal-based zero-divisor graph ΓI (R), and the compressed zero-divisor graphs ΓE(R) and ΓE(R/I). In this paper, we investigate the structure and relationship between the various ∼-zero-divisor graphs.

Dominating sets of the comaximal and ideal-based zero-divisor graphs of commutative rings

Let R be a commutative ring with nonzero identity, and let I be an ideal of R. The ideal-based zero-divisor graph of R, denoted by ΓI (R), is the graph whose vertices are the set {x ∈ R \\ I| xy ∈ I for some y∈ R \\ I} and two distinct vertices x and y are adjacent if and only if xy ∈ I. Define the comaximal graph of R, denoted by CG(R), to be a graph whose vertices are the elements of R, where two distinct vertices a and b are adjacent if and only if Ra+Rb=R. A nonempty set S ⊆ V of a graph G=(V, E) is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this paper is to study the dominating sets and domination number of ΓI (R) and the comaximal graph CG2(R) \\ J (R) (or CGJ (R) for short) where CG2(R) is the subgraph of CG(R) induced on the nonunit elements of R and J (R) is the Jacobson radical of R.

A generalized ideal-based zero-divisor graph

Let R be a commutative ring with identity, I its proper ideal and M be a unitary R-module. In this paper, we introduce and study a kind of graph structure of an R-module M with respect to proper ideal I, denoted by ΓI(RM) or simply ΓI(M). It is the (undirected) graph with the vertex set M\{0} and two distinct vertices x and y are adjacent if and only if [x : M][y : M] ⊆ I. Clearly, the zero-divisor graph of R is a subgraph of Γ0(R); this is an important result on the definition. We prove that if ann R(M) ⊆ I and H is the subgraph of ΓI(M) induced by the set of all non-isolated vertices, then diam (H) ≤ 3 and gr (ΓI(M)) ∈ {3, 4, ∞}. Also, we prove that if Spec (R) and ω(Γ Nil (R)(M)) are finite, then χ(Γ Nil (R)(M)) ≤ ∣ Spec (R)∣ + ω(Γ Nil (R)(M)). Moreover, for a secondary R-module M and prime ideal P, we determine the chromatic number and the clique number of ΓP(M), where ann R(M) ⊆ P. Among other results, it is proved that for a semisimple R-module M with ann R(M) ⊆ I, ΓI(M) is a forest if and only if ΓI(M) is a union of isolated vertices or a star.

On Some Graphs Associated to Commutative Semirings

Similar to the case of commutative rings, we define the notion of a zero-divisor graph of a commutative semiring R with respect to an ideal I of R, denoted by \({\Gamma_I(R)}\). That is an undirected graph whose vertex set is the set \({\{x \in R {\setminus} I | xy \in I {\rm for some} y \in R {\setminus} I\}}\) with distinct vertices x and y adjacent if and only if xy ∈ I. We discuss when \({\Gamma_I(R)}\) is r-partite. We also give some results on the subgraphs and the parameters of \({\Gamma_I(R)}\). In addition, we apply these results to the annihilating-ideal graph of R with respect to I, denoted by \({\mathbb{AG}_I(R)}\), as an example (special case) of \({\Gamma_I(R)}\). It is also shown that for I a radical ideal of \({R, \Gamma_I(R)}\) is isomorphic to a subgraph of \({\mathbb{AG}_I(R)}\) which naturally asserts some known results (properties) between them interchangeably. Finally, we provide some counterexamples and construct a semiring whose ideal-based zero-divisor graph has a cut-point and more than one bridge, which is in contrast to the ring case.

The ideal-based zero-divisor graph of commutative chained rings

Let I be a proper ideal of a commutative ring R with 1 6 0. The ideal-based zero-divisor graph of R with respect to I, denoted by I(R), is the (simple) graph with vertices {x 2 R I | xy 2 I for some y 2 R I }, and distinct vertices x and y are adjacent if and only if xy 2 I. In this paper, we study I(R) for commutative rings R such that R/I is a chained ring.

An ideal-based zero-divisor graph of direct products of commutative rings

In this paper, specifically, we look at the preservation of the diameter and girth of the zero-divisor graph with respect to an ideal of a commutative ring when extending to a finite direct product of commutative rings.

Some Remarks on the Graph Γ I (R)

Let R be a commutative ring with nonzero identity and Z(R) its set of zero-divisors. The zero-divisor graph of R is Γ(R), with vertices Z(R)∖{0} and distinct vertices x and y are adjacent if and only if xy = 0. For a proper ideal I of R, the ideal-based zero-divisor graph of R is Γ I (R), with vertices {x ∈ R∖I | xy ∈ I for some y ∈ R∖I} and distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we study the relationship between the two graphs Γ(R) and Γ I (R). We also determine when Γ I (R) is either a complete graph or a complete bipartite graph and investigate when Γ I (R) ≅ Γ(S) for some commutative ring S.

Annihilator Ideal-Based Zero-Divisor Graphs Over Multiplication Modules

For a commutative ring R with identity, an ideal-based zero-divisor graph, denoted by Γ I (R), is the graph whose vertices are {x ∈ R∖I | xy ∈ I for some y ∈ R∖I}, and two distinct vertices x and y are adjacent if and only if xy ∈ I. In this article, we investigate an annihilator ideal-based zero-divisor graph by replacing the ideal I with the annihilator ideal Ann(M) for a multiplication R-module M. Based on the above-mentioned definition, we examine some properties of an R-module over a von Neumann regular ring, and the cardinality of an R-module associated with Γ Ann(M)(R).

Dominating Sets of Some Graphs Associated to Commutative Rings

Let G = (V, E) be a graph. A set S ⊆ V is a dominating set of G if every vertex in V is either in S or is adjacent to a vertex in S. The domination number γ(G) of G is the minimum cardinality among the dominating sets of G. The main object of this article is to study and characterize the dominating sets of the zero-divisor graph Γ(R) and ideal-based zero-divisor graph Γ I (R) of a commutative ring R.

On the diameter of the graph ГAnn(M)(R)

For a commutative ring R with identity, the ideal-based zero-divisor graph, denoted by ?I (R), is the graph whose vertices are {x ? R\I|xy ? I for some y ? R\I}, and two distinct vertices x and y are adjacent if and only if xy?I. In this paper, we investigate an annihilator ideal-based zero-divisor graph, denoted by ?Ann(M)(R), by replacing the ideal I with the annihilator ideal Ann(M) for an R-module M. We also study the relationship between the diameter of ?Ann(M) (R) and the minimal prime ideals of Ann(M). In addition, we determine when ?Ann(M)(R) is complete. In particular, we prove that for a reduced R-module M, ?Ann(M) (R) is a complete graph if and only if R ? Z2?Z2 and M ? M1?M2 for M1 and M2 nonzero Z2-modules.

Zero-Divisor Graphs with respect to Ideals in Noncommutative Rings

Let R be a commutative ring and I an ideal of R. The zero-divisor graph of R with respect to I, denoted ΓI(R), is the undirected graph whose vertex set is {x∈R∖I|xy∈I for some y∈R∖I} with two distinct vertices x and y joined by an edge when xy∈I. In this paper, we extend the definition of the ideal-based zero-divisor graph to noncommutative rings.

A GENERALIZED IDEAL BASED-ZERO DIVISOR GRAPHS OF NEAR-RINGS

In this paper, we introduce the generalized ideal-based zero-divisor graph structure of near-ring N, denoted by <TEX>$\widehat{{\Gamma}_I(N)}$</TEX>. It is shown that if I is a completely reflexive ideal of N, then every two vertices in <TEX>$\widehat{{\Gamma}_I(N)}$</TEX> are connected by a path of length at most 3, and if <TEX>$\widehat{{\Gamma}_I(N)}$</TEX> contains a cycle, then the core K of <TEX>$\widehat{{\Gamma}_I(N)}$</TEX> is a union of triangles and rectangles. We have shown that if <TEX>$\widehat{{\Gamma}_I(N)}$</TEX> is a bipartite graph for a completely semiprime ideal I of N, then N has two prime ideals whose intersection is I.