scholarly journals On Certain Bounds for Edge Metric Dimension of Zero-Divisor Graphs Associated with Rings

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Hafiz Muahmmad Afzal Siddiqui ◽  
Ammar Mujahid ◽  
Muhammad Ahsan Binyamin ◽  
Muhammad Faisal Nadeem
Keyword(s):  
Zero Divisor ◽  
Research Work ◽  
Commutative Rings ◽  
Polynomial Rings ◽  
Metric Dimension ◽  
Gaussian Integers ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Algebraic Properties ◽  
Do So

Given a finite commutative unital ring S having some non-zero elements x ,   y such that x . y = 0 , the elements of S that possess such property are called the zero divisors, denoted by Z S . We can associate a graph to S with the help of zero-divisor set Z S , denoted by ζ S (called the zero-divisor graph), to study the algebraic properties of the ring S . In this research work, we aim to produce some general bounds for the edge version of metric dimension regarding zero-divisor graphs of S . To do so, we will discuss the zero-divisor graphs for the ring of integers ℤ m modulo m , some quotient polynomial rings, and the ring of Gaussian integers ℤ m i modulo m . Then, we prove the general result for the bounds of edge metric dimension of zero-divisor graphs in terms of maximum degree and diameter of ζ S . In the end, we provide the commutative rings with the same metric dimension, edge metric dimension, and upper dimension.

scholarly journals Metric and upper dimension of zero divisor graphs associated to commutative rings

2020 ◽  
Vol 12 (1) ◽  
pp. 84-101 ◽  
Author(s):  
S. Pirzada ◽  
M. Aijaz
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Metric Dimension ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Finite Commutative Rings

AbstractLet R be a commutative ring with Z*(R) as the set of non-zero zero divisors. The zero divisor graph of R, denoted by Γ(R), is the graph whose vertex set is Z*(R), where two distinct vertices x and y are adjacent if and only if xy = 0. In this paper, we investigate the metric dimension dim(Γ(R)) and upper dimension dim+(Γ(R)) of zero divisor graphs of commutative rings. For zero divisor graphs Γ(R) associated to finite commutative rings R with unity 1 ≠ 0, we conjecture that dim+(Γ(R)) = dim(Γ(R)), with one exception that {\rm{R}} \cong \Pi {\rm\mathbb{Z}}_2^{\rm{n}}, n ≥ 4. We prove that this conjecture is true for several classes of rings. We also provide combinatorial formulae for computing the metric and upper dimension of zero divisor graphs of certain classes of commutative rings besides giving bounds for the upper dimension of zero divisor graphs of rings.

Characterizations of Three Classes of Zero-Divisor Graphs

2012 ◽  
Vol 55 (1) ◽  
pp. 127-137 ◽  
Author(s):  
John D. LaGrange
Keyword(s):  
Commutative Ring ◽  
Direct Product ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Central Vertex ◽  
Integral Domains ◽  
Boolean Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

AbstractThe zero-divisor graph Γ(R) of a commutative ring R is the graph whose vertices consist of the nonzero zero-divisors of R such that distinct vertices x and y are adjacent if and only if xy = 0. In this paper, a characterization is provided for zero-divisor graphs of Boolean rings. Also, commutative rings R such that Γ(R) is isomorphic to the zero-divisor graph of a direct product of integral domains are classified, as well as those whose zero-divisor graphs are central vertex complete.

scholarly journals Planar, Outerplanar, and Toroidal Graphs of the Generalized Zero-Divisor Graph of Commutative Rings

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Abdulaziz M. Alanazi ◽  
Mohd Nazim ◽  
Nadeem Ur Rehman
Keyword(s):  
Zero Divisor ◽  
Domination Number ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Nonzero Intersection ◽  
Toroidal Graphs ◽  
Vertex Set ◽  
Generalized Zero ◽  
Finite Commutative Rings

Let A be a commutative ring with unity and let set of all zero divisors of A be denoted by Z A . An ideal ℐ of the ring A is said to be essential if it has a nonzero intersection with every nonzero ideal of A . It is denoted by ℐ ≤ e A . The generalized zero-divisor graph denoted by Γ g A is an undirected graph with vertex set Z A ∗ (set of all nonzero zero-divisors of A ) and two distinct vertices x 1 and x 2 are adjacent if and only if ann x 1 + ann x 2 ≤ e A . In this paper, first we characterized all the finite commutative rings A for which Γ g A is isomorphic to some well-known graphs. Then, we classify all the finite commutative rings A for which Γ g A is planar, outerplanar, or toroidal. Finally, we discuss about the domination number of Γ g A .

The maximal degree of a zero-divisor graph

2019 ◽  
pp. 2050151
Author(s):  
Husam Q. Mohammad ◽  
Nazar H. Shuker ◽  
Luma A. Khaleel
Keyword(s):  
Maximum Degree ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Finite Ring ◽  
Maximal Degree ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Finite Commutative Rings

The rings considered in this paper are finite commutative rings with identity, which are not fields. For any ring [Formula: see text] which is not a field and which is not necessarily finite, we denote the set of all zero-divisors of [Formula: see text] by [Formula: see text] and [Formula: see text] by [Formula: see text]. Let [Formula: see text] denote the zero-divisor graph of [Formula: see text] and for a finite ring [Formula: see text], let [Formula: see text] denote the maximum degree of [Formula: see text]. We denote [Formula: see text] by [Formula: see text]. The aim of this paper is to study some properties of [Formula: see text].

scholarly journals Eigenvalues of zero-divisor graphs of finite commutative rings

2020 ◽  
Author(s):  
Katja Mönius
Keyword(s):  
Prime Number ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Graph Product ◽  
Graph Theoretic ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Finite Commutative Rings

AbstractWe investigate eigenvalues of the zero-divisor graph $$\Gamma (R)$$ Γ ( R ) of finite commutative rings R and study the interplay between these eigenvalues, the ring-theoretic properties of R and the graph-theoretic properties of $$\Gamma (R)$$ Γ ( R ) . The graph $$\Gamma (R)$$ Γ ( R ) is defined as the graph with vertex set consisting of all nonzero zero-divisors of R and adjacent vertices x, y whenever $$xy = 0$$ x y = 0 . We provide formulas for the nullity of $$\Gamma (R)$$ Γ ( R ) , i.e., the multiplicity of the eigenvalue 0 of $$\Gamma (R)$$ Γ ( R ) . Moreover, we precisely determine the spectra of $$\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)$$ Γ ( Z p × Z p × Z p ) and $$\Gamma ({\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p \times {\mathbb {Z}}_p)$$ Γ ( Z p × Z p × Z p × Z p ) for a prime number p. We introduce a graph product $$\times _{\Gamma }$$ × Γ with the property that $$\Gamma (R) \cong \Gamma (R_1) \times _{\Gamma } \cdots \times _{\Gamma } \Gamma (R_r)$$ Γ ( R ) ≅ Γ ( R 1 ) × Γ ⋯ × Γ Γ ( R r ) whenever $$R \cong R_1 \times \cdots \times R_r.$$ R ≅ R 1 × ⋯ × R r . With this product, we find relations between the number of vertices of the zero-divisor graph $$\Gamma (R)$$ Γ ( R ) , the compressed zero-divisor graph, the structure of the ring R and the eigenvalues of $$\Gamma (R)$$ Γ ( R ) .

On zero-divisor graphs of skew polynomial rings over non-commutative rings

2017 ◽  
Vol 16 (03) ◽  
pp. 1750056 ◽  
Author(s):  
E. Hashemi ◽  
R. Amirjan ◽  
A. Alhevaz
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Polynomial Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Skew Polynomial Rings ◽  
Derivation Formula ◽  
Base Ring ◽  
Skew Polynomial ◽  
Associative Rings

In this paper, we continue to study zero-divisor properties of skew polynomial rings [Formula: see text], where [Formula: see text] is an associative ring equipped with an endomorphism [Formula: see text] and an [Formula: see text]-derivation [Formula: see text]. For an associative ring [Formula: see text], the undirected zero-divisor graph of [Formula: see text] is the graph [Formula: see text] such that the vertices of [Formula: see text] are all the nonzero zero-divisors of [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are connected by an edge if and only if [Formula: see text] or [Formula: see text]. As an application of reversible rings, we investigate the interplay between the ring-theoretical properties of a skew polynomial ring [Formula: see text] and the graph-theoretical properties of its zero-divisor graph [Formula: see text]. Our goal in this paper is to give a characterization of the possible diameters of [Formula: see text] in terms of the diameter of [Formula: see text], when the base ring [Formula: see text] is reversible and also have the [Formula: see text]-compatible property. We also completely describe the associative rings all of whose zero-divisor graphs of skew polynomials are complete.

On diameter of the zero-divisor and the compressed zero-divisor graphs of skew Laurent polynomial rings

2019 ◽  
Vol 18 (07) ◽  
pp. 1950126 ◽  
Author(s):  
Ebrahim Hashemi ◽  
Mona Abdi ◽  
Abdollah Alhevaz
Keyword(s):  
Zero Divisor ◽  
Undirected Graph ◽  
Complete Characterization ◽  
Laurent Polynomial ◽  
Polynomial Rings ◽  
Noncommutative Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Base Ring

Let [Formula: see text] be an associative ring with nonzero identity. The zero-divisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the nonzero zero-divisors of [Formula: see text], and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. Let [Formula: see text] and [Formula: see text] be the set of all right annihilators and the set of all left annihilator of an element [Formula: see text], respectively, and let [Formula: see text]. The relation on [Formula: see text] given by [Formula: see text] if and only if [Formula: see text] is an equivalence relation. The compressed zero-divisor graph [Formula: see text] of [Formula: see text] is the (undirected) graph with vertices the equivalence classes induced by [Formula: see text] other than [Formula: see text] and [Formula: see text], and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] or [Formula: see text]. The goal of our paper is to study the diameter of zero-divisor and the compressed zero-divisor graph of skew Laurent polynomial rings over noncommutative rings. We give a complete characterization of the possible diameters of [Formula: see text] and [Formula: see text], where the base ring [Formula: see text] is reversible and also has the [Formula: see text]-compatible property.

Locating sets and numbers of graphs associated to commutative rings

2014 ◽  
Vol 13 (07) ◽  
pp. 1450047 ◽  
Author(s):  
S. Pirzada ◽  
Rameez Raja ◽  
Shane Redmond
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Undirected Graph ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Finite Vector ◽  
Minimum Number

For a graph G(V, E) with order n ≥ 2, the locating code of a vertex v is a finite vector representing distances of v with respect to vertices of some ordered subset W of V(G). The set W is a locating set of G(V, E) if distinct vertices have distinct codes. A locating set containing a minimum number of vertices is a minimum locating set for G(V, E). The locating number denoted by loc (G) is the number of vertices in the minimum locating set. Let R be a commutative ring with identity 1 ≠ 0, the zero-divisor graph denoted by Γ(R), is the (undirected) graph whose vertices are the nonzero zero-divisors of R with two distinct vertices joined by an edge when the product of vertices is zero. We introduce and investigate locating numbers in zero-divisor graphs of a commutative ring R. We then extend our definition to study and characterize the locating numbers of an ideal based zero-divisor graph of a commutative ring R.

scholarly journals On planarity of compressed zero-divisor graphs associated to commutative rings

2020 ◽  
Vol 29 (2) ◽  
pp. 131-136
Author(s):  
M. IMRAN BHAT ◽  
S. PIRZADA ◽  
AHMAD M. ALGHAMDI
Keyword(s):  
Equivalence Class ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Equivalence Classes ◽  
Local Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set

The equivalence class [r] of an element r ∈ R is the set of zero-divisors s such that ann(r) = ann(s), that is, [r] = {s ∈ R : ann(r) = ann(s). The compressed zero-divisor graph, denoted by Γc(R), is the compression of a zero-divisor graph, in which the vertex set is the set of all equivalence classes of nonzero zero-divisors of a ring R, that is, the vertex set of Γc(R) is Re − {[0], [1]}, where Re = {[r] : r ∈ R} and two distinct equivalence classes [r] and [s] are adjacent if and only if rs = 0. In this article, we investigate the planarity of Γc(R) for some finite local rings of order p 2 , p 3 and determine the planarity of compressed zero-divisor graph of some local rings of order 32, whose zero-divisor graph is nonplanar. Further, we determine values of m and n for which Γc(Zn) and Γc(Zn[x]/(xm)) are planar.

ON RINGS R WHOSE GRAPHS Γ(R) SATISFY CONDITION (Kp)

2011 ◽  
Vol 10 (04) ◽  
pp. 665-674
Author(s):  
LI CHEN ◽  
TONGSUO WU
Keyword(s):  
Prime Number ◽  
Commutative Ring ◽  
Complete Graph ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Satisfy Condition ◽  
Induced Subgraph ◽  
Zero Divisor Graph ◽  
Ring Graph ◽  
Finite Commutative Rings

Let p be a prime number. Let G = Γ(R) be a ring graph, i.e. the zero-divisor graph of a commutative ring R. For an induced subgraph H of G, let CG(H) = {z ∈ V(G) ∣N(z) = V(H)}. Assume that in the graph G there exists an induced subgraph H which is isomorphic to the complete graph Kp-1, a vertex c ∈ CG(H), and a vertex z such that d(c, z) = 3. In this paper, we characterize the finite commutative rings R whose graphs G = Γ(R) have this property (called condition (Kp)).

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