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.

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 ) .

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 Thek-Zero-Divisor Hypergraph of a Commutative Ring

2007 ◽  
Vol 2007 ◽  
pp. 1-15 ◽  
Author(s):  
Ch. Eslahchi ◽  
A. M. Rahimi
Keyword(s):  
Commutative Ring ◽  
Nilpotent Element ◽  
Zero Divisor ◽  
Clique Number ◽  
Common Point ◽  
Prime Ideals ◽  
Uniform Hypergraph ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set

The concept of the zero-divisor graph of a commutative ring has been studied by many authors, and thek-zero-divisor hypergraph of a commutative ring is a nice abstraction of this concept. Though some of the proofs in this paper are long and detailed, any reader familiar with zero-divisors will be able to read through the exposition and find many of the results quite interesting. LetRbe a commutative ring andkan integer strictly larger than2. Ak-uniform hypergraphHk(R)with the vertex setZ(R,k), the set of allk-zero-divisors inR, is associated toR, where eachk-subset ofZ(R,k)that satisfies thek-zero-divisor condition is an edge inHk(R). It is shown that ifRhas two prime idealsP1andP2with zero their only common point, thenHk(R)is a bipartite (2-colorable) hypergraph with partition setsP1−Z′andP2−Z′, whereZ′is the set of all zero divisors ofRwhich are notk-zero-divisors inR. IfRhas a nonzero nilpotent element, then a lower bound for the clique number ofH3(R)is found. Also, we have shown thatH3(R)is connected with diameter at most 4 wheneverx2≠0for all3-zero-divisorsxofR. Finally, it is shown that for any finite nonlocal ringR, the hypergraphH3(R)is complete if and only ifRis isomorphic toZ2×Z2×Z2.

scholarly journals On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 482
Author(s):  
Bilal A. Rather ◽  
Shariefuddin Pirzada ◽  
Tariq A. Naikoo ◽  
Yilun Shang
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Simple Graph ◽  
Laplacian Spectrum ◽  
Laplacian Eigenvalues ◽  
Ring Of Integers ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Positive Integers

Given a commutative ring R with identity 1≠0, let the set Z(R) denote the set of zero-divisors and let Z*(R)=Z(R)∖{0} be the set of non-zero zero-divisors of R. The zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is Z*(R) and each pair of vertices in Z*(R) are adjacent when their product is 0. In this article, we find the structure and Laplacian spectrum of the zero-divisor graphs Γ(Zn) for n=pN1qN2, where p<q are primes and N1,N2 are positive integers.

scholarly journals Distance-Based Topological Polynomials Associated with Zero-Divisor Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ali Ahmad ◽  
S. C. López
Keyword(s):  
Commutative Ring ◽  
Zero Divisor ◽  
Wiener Index ◽  
Formula One ◽  
Zero Divisor Graph ◽  
Zero Divisors ◽  
Vertex Set ◽  
Hosoya Polynomial ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity and let Z R be its set of zero divisors. The zero-divisor graph of R is the graph Γ R with vertex set V Γ R = Z R ∗ , where Z R ∗ = Z R \ 0 , and edge set E Γ R = x , y :   x ⋅ y = 0 . One of the basic results for these graphs is that Γ R is connected with diameter less than or equal to 3. In this paper, we obtain a few distance-based topological polynomials and indices of zero-divisor graph when the commutative ring is ℤ p 2 q 2 , namely, the Wiener index, the Hosoya polynomial, and the Shultz and the modified Shultz indices and polynomials.

Ideal-Based k-Zero-Divisor Hypergraph of Commutative Rings

2021 ◽  
Vol 28 (04) ◽  
pp. 655-672
Author(s):  
K. Selvakumar ◽  
M. Subajini
Keyword(s):  
Commutative Ring ◽  
Finite Fields ◽  
Zero Divisor ◽  
Necessary Conditions ◽  
Commutative Rings ◽  
Nonzero Ideal ◽  
Fixed Integer ◽  
Zero Divisors ◽  
Vertex Set ◽  
The Ideal

Let [Formula: see text] be a commutative ring, [Formula: see text] an ideal of [Formula: see text] and [Formula: see text] a fixed integer. The ideal-based [Formula: see text]-zero-divisor hypergraph [Formula: see text] of [Formula: see text] has vertex set [Formula: see text], the set of all ideal-based [Formula: see text]-zero-divisors of [Formula: see text], and for distinct elements [Formula: see text] in [Formula: see text], the set [Formula: see text] is an edge in [Formula: see text] if and only if [Formula: see text] and the product of the elements of any [Formula: see text]-subset of [Formula: see text] is not in [Formula: see text]. In this paper, we show that [Formula: see text] is connected with diameter at most 4 provided that [Formula: see text] for all ideal-based 3-zero-divisor hypergraphs. Moreover, we find the chromatic number of [Formula: see text] when [Formula: see text] is a product of finite fields. Finally, we find some necessary conditions for a finite ring [Formula: see text] and a nonzero ideal [Formula: see text] of [Formula: see text] to have [Formula: see text] planar.

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].

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