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 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.

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 .

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 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)).

On zero-divisor graphs of commutative rings without identity

2019 ◽  
Vol 19 (12) ◽  
pp. 2050226 ◽  
Author(s):  
G. Kalaimurugan ◽  
P. Vignesh ◽  
T. Tamizh Chelvam
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Ring ◽  
Free Order ◽  
Finite Commutative Rings

Let [Formula: see text] be a finite commutative ring without identity. In this paper, we characterize all finite commutative rings without identity, whose zero-divisor graphs are unicyclic, claw-free and tree. Also, we obtain all finite commutative rings without identity and of cube-free order for which the corresponding zero-divisor graph is toroidal.

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 ON VARIETIES OF RINGS WHOSE FINITE RINGS ARE DETERMINED BY THEIR ZERO-DIVISOR GRAPHS

2012 ◽  
Vol 05 (02) ◽  
pp. 1250019 ◽  
Author(s):  
A. S. Kuzmina ◽  
Yu. N. Maltsev
Keyword(s):  
Associative Ring ◽  
Zero Divisor ◽  
Finite Ring ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Zero Divisors

The zero-divisor graph Γ(R) of an associative ring R is the graph whose vertices are all nonzero zero-divisors (one-sided and two-sided) of R, and two distinct vertices x and y are joined by an edge if and only if either xy = 0 or yx = 0. In the present paper, we study some properties of ring varieties where every finite ring is uniquely determined by its zero-divisor graph.

Finite Rings Whose Graphs Have Clique Number Less than Five

2021 ◽  
Vol 28 (03) ◽  
pp. 533-540
Author(s):  
Qiong Liu ◽  
Tongsuo Wu ◽  
Jin Guo
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Clique Number ◽  
Commutative Rings ◽  
Finite Rings ◽  
Zero Divisor Graph ◽  
Finite Commutative Rings

Let [Formula: see text] be a commutative ring and [Formula: see text] be its zero-divisor graph. We completely determine the structure of all finite commutative rings whose zero-divisor graphs have clique number one, two, or three. Furthermore, if [Formula: see text] (each [Formula: see text] is local for [Formula: see text]), we also give algebraic characterizations of the ring [Formula: see text] when the clique number of [Formula: see text] is four.

scholarly journals Categorial properties of compressed zero-divisor graphs of finite commutative rings

2020 ◽  
pp. 2150069
Author(s):  
Alen Đurić ◽  
Sara Jevđnić ◽  
Nik Stopar
Keyword(s):  
Principal Ideal ◽  
Zero Divisor ◽  
Commutative Rings ◽  
Local Rings ◽  
Unital Ring ◽  
Zero Divisor Graph ◽  
Definition Of ◽  
Unital Rings ◽  
Finite Commutative Rings

By modifying the existing definition of a compressed zero-divisor graph [Formula: see text], we define a compressed zero-divisor graph [Formula: see text] of a finite commutative unital ring [Formula: see text], where the compression is performed by means of the associatedness relation (a refinement of the relation used in the definition of [Formula: see text]). We prove that this is the best possible compression which induces a functor [Formula: see text], and that this functor preserves categorial products (in both directions). We use the structure of [Formula: see text] to characterize important classes of finite commutative unital rings, such as local rings and principal ideal rings.

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.

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