Zero-divisor Graph Research Articles

Chromatic uniqueness of zero-divisor graphs

The zero-divisor graph Π(R) of a commutative ring R is the graph whose vertices are the elements of R such that the vertices u and v are adjacent if and only if uv = 0. If the graphs G and H have the same chromatic polynomial, then we say that they are chromatically equivalent (or χ−equivalent), written as G ∼ H. Suppose a graph is uniquely determined by its chromatic polynomial. Then it is said to be chromatically unique (or χ-unique). In this paper, we discuss the question: For which numbers n is the graph Π(Zn) χ-unique? While Zn is one of the simplest rings, we proved that for any graph A0, for some n, Π(Zn) contains an induced subgraph isomorphic to A0. The first result in the subject states that for n ≥ 10 even, Π(Zn) is not χ-unique (Gehet, Khalaf). By definition, n is square-free if it is prime or the product of different prime numbers. Our main result is the following. If n ≥ 10 is neither square-free nor the square of a prime then it is not χ-unique. Here and in our preceding work, we use a common method. For odd square-free non-prime n, the problem is open, though on the structure of Π(Zn) we know much in this case.

Decomposition of zero divisor graph into cycles and stars

For a graph G and a subgraph H of G, an H-decomposition of G is a partition of the edge set of G into subsets Ei,1⩽i⩽k, such that each Ei induces a graph isomorphic to H. A graph Γ(ℝ) is said to be non-zero zero divisor graph of commutative ring ℝ with identity if u,v∈V(Γ(ℝ)) and (u,v)∈E(Γ(ℝ)) if and only if uv=0. It is prove that complete decomposible into cycle of length 4 of an H-decomposition of the zero divisor graph Γ(ℝ) where H is any simple connected graph. In particular, we give a complete solution to the problem in the case Zp×Zp×Zp×,…,×Zp (n times). For any positive integer n>2, there exists a decomposition of Γ(ℝ) into cycle and stars in a commutative ring ℝ. We show that the obvious the graph Γ(ℝ) is decomposition into cycle and stars. Overall, the proposed of the graph Γ(ℝ) has significantly improved the decomposing to algebraic structure which can be useful for networking. In this paper we investigate the concept of Γ(ℝ) is decomposition into cycles and stars as a commutative rings R=Zp×Zp,Zp×Zp×Zp,Zp×Zp×Zp×Zp and Zp×Zp×Zp×,…,×Zp with p is a prime number. It is prove that the zero divisor graph Γ(R) is complete decomposible into cycle of length 4 and star. In particular, we give a complete solution to the problem in the case Zp×Zp×Zp×,…,×Zp (n times). For any positive integer n>2, there exists a decomposition of Γ(R) into cycle and stars in a commutative ring ℝ. We show that the obvious the graph Γ(R) is decomposition into cycle and stars. Overall, the proposed of the graph Γ(R) has significantly improved the decomposing to algebraic structure which can be useful for networking area.

Properties of derived signed graphs on Beck’s zero-divisor graph

The concept of associating a graph to a ring was initiated by Beck in 1988. The zero-divisor graph of a commutative ring [Formula: see text] with unity ([Formula: see text]) is defined as a simple graph, denoted by [Formula: see text], where elements of the ring [Formula: see text] represent vertices and two distinct vertices in [Formula: see text] are adjacent if the product of the corresponding elements is zero in [Formula: see text]. Here, we extend the notion of zero-divisor graphs to signed graphs. We introduce four types of signing to the zero-divisor graphs. Further, we characterize rings for which these four types of signed graphs, its lined signed graphs and their negations are balanced, clusterable, sign-compatible, canonically sign-compatible, and canonically consistent.

The Metric Chromatic Number of Zero Divisor Graph of a Ring Z n

Let Γ be a nontrivial connected graph, c : V Γ ⟶ ℕ be a vertex colouring of Γ , and L i be the colouring classes that resulted, where i = 1,2 , … , k . A metric colour code for a vertex a of a graph Γ is c a = d a , L 1 , d a , L 2 , … , d a , L n , where d a , L i is the minimum distance between vertex a and vertex b in L i . If c a ≠ c b , for any adjacent vertices a and b of Γ , then c is called a metric colouring of Γ as well as the smallest number k satisfies this definition which is said to be the metric chromatic number of a graph Γ and symbolized μ Γ . In this work, we investigated a metric colouring of a graph Γ Z n and found the metric chromatic number of this graph, where Γ Z n is the zero-divisor graph of ring Z n .

Prime Decomposition of Zero Divisor Graph in a Commutative Ring

Let R be a commutative ring and let Γ Z n be the zero divisor graph of a commutative ring R , whose vertices are nonzero zero divisors of Z n , and such that the two vertices u , v are adjacent if n divides u v . In this paper, we introduce the concept of prime decomposition of zero divisor graph in a commutative ring and also discuss some special cases of Γ Z 3 p , Γ Z 5 p , Γ Z 7 p , and Γ Z p q .

Computing Wiener and Hyper-Wiener Indices of Zero-Divisor Graph of ℤ ℊ 3 × ℤ I 1 I 2

Let S = ℤ ℊ 3 × ℤ I 1 I 2 be a commutative ring where ℊ , I 1 and I 2 are positive prime integers with I 1 ≠ I 2 . The zero-divisor graph assigned to S is an undirected graph, denoted as Y S with vertex set V( Y (S)) consisting of all Zero-divisor of the ring S and for any c, d ∈ V( Y (S)), c d ∈ E Y S if and only if cd =0. A topological index/descriptor is described as a topological-invariant quantity that transforms a molecular graph into a mathematical real number. In this paper, we have computed distance-based polynomials of Y R i-e Hosoya polynomial, Harary polynomial, and the topological indices related to these polynomials namely Wiener index, and Hyper-Wiener index.

The wiener index of the zero-divisor graph for a new class of residue class rings

The zero-divisor graph of a commutative ring R, denoted by Γ(R), is a graph whose two distinct vertices x and y are joined by an edge if and only if xy = 0 or yx = 0. The main problem of the study of graphs defined on algebraic structure is to recognize finite rings through the properties of various graphs defined on it. The main objective of this article is to study the Wiener index of zero-divisor graph and compressed zero-divisor graph of the ring of integer modulo p s q t for all distinct primes p, q and . We study the structure of these graphs by dividing the vertex set. Furthermore, a formula for the Wiener index of zero-divisor graph of Γ(R), and a formula for the Wiener index of associated compressed zero-divisor graph Γ E (R) are derived for .

On the linear strand of edge ideals of some zero-divisor graphs

Let I(G) be the edge ideal associated to a simple graph G. Given a prime number p and an integer, we study the -graded Betti numbers that appear in the linear strand of the minimal free resolution of , where is the zero-divisor graph of the ring . In particular, we compute the extremal Betti number of . As a consequence, we find the Castelnuovo-Mumford regularity and projective dimension of these ideals. Moreover, we exhibit formulae that determine all -graded Betti numbers in the linear strand of the minimal free resolution of for certain values of n.

Universal Adjacency Spectrum of the Looped Zero Divisor Graph of $$\mathbb {Z}_n$$

Universal Adjacency Spectrum of the Looped Zero Divisor Graph of $$\mathbb {Z}_n$$

On graphs associated to ring of Guassian integers and ring of integers modulo n

Abstract For a commutative ring R with identity 1, the zero-divisor graph of R, denoted by Γ(R), is a simple graph whose vertex set is the set of non-zero zero divisors Z*(R) and the two vertices x and y ∈ Z*(R) are adjacent if and only if xy = 0. In this paper, we compute the values of some graph parameters of the zero-divisor graph associated to the ring of Gaussian integers modulo n, ℤn[i] and the ring of integers modulo n, ℤn.

Annihilator graphs of a commutative semigroup whose Zero-divisor graphs are a complete graph with an end vertex

Abstract Suppose that the zero-divisor graph of a commutative semi-group S, be a complete graph with an end vertex. In this paper, we determine the structure of the annihilator graph S and we show that if Z(S)= S, then the annihilator graph S is a disconnected graph.

Some Properties of the Zero-Divisor Graphs of Idealization Ring R(+)M

The aim of this article to follow the properties of the zero-divisor graph of special idealization ring. We study the wiener index of the zero-divisors graph of some special idealization ring R(+)M and find the clique number of the graph Γ(R(+)M) is ω (Γ(R(+)M)) = |M| − 1, where R is an integral domain. We also discuss when the zero-divisors graph of some special idealization ring R(+)M are Hamiltonian graph.

On normalized distance Laplacian eigenvalues of graphs and applications to graphs defined on groups and rings

The normalized distance Laplacian matrix of a connected graph $ G $, denoted by $ D^{\mathcal{L}}(G) $, is defined by $ D^{\mathcal{L}}(G)=Tr(G)^{-1/2}D^L(G)Tr(G)^{-1/2}, $ where $ D(G) $ is the distance matrix, the $D^{L}(G)$ is the distance Laplacian matrix and $ Tr(G)$ is the diagonal matrix of vertex transmissions of $ G. $ The set of all eigenvalues of $ D^{\mathcal{L}}(G) $ including their multiplicities is the normalized distance Laplacian spectrum or $ D^{\mathcal{L}} $-spectrum of $G$. In this paper, we find the $ D^{\mathcal{L}} $-spectrum of the joined union of regular graphs in terms of the adjacency spectrum and the spectrum of an auxiliary matrix. As applications, we determine the $ D^{\mathcal{L}} $-spectrum of the graphs associated with algebraic structures. In particular, we find the $ D^{\mathcal{L}} $-spectrum of the power graphs of groups, the $ D^{\mathcal{L}} $-spectrum of the commuting graphs of non-abelian groups and the $ D^{\mathcal{L}} $-spectrum of the zero-divisor graphs of commutative rings. Several open problems are given for further work.

Nearring of ore extensions: Zero-divisors and regular elements

Let [Formula: see text] be a symmetric and [Formula: see text]-compatible ring. In this paper, we first specify the zero-divisor elements of the near-ring [Formula: see text]. We prove that if [Formula: see text] is an [Formula: see text]-rigid ring, then the set of all zero-divisor elements of the near-ring [Formula: see text] forms an ideal of [Formula: see text] if and only if [Formula: see text] is an ideal of [Formula: see text] and [Formula: see text] has the right Property [Formula: see text]. Also, we are interested in studying the zero-divisor graph of [Formula: see text] which is denoted by [Formula: see text]. It is shown that [Formula: see text], and if [Formula: see text] is not reduced, then [Formula: see text] for each [Formula: see text] if and only if [Formula: see text]. Moreover, we characterize the units, the clean elements, the regular elements, the nilpotent elements and also the [Formula: see text]-regular elements of the near-ring [Formula: see text], where [Formula: see text] is a reversible and [Formula: see text]-compatible ring and [Formula: see text] is a countable locally nilpotent ideal of [Formula: see text]. Finally, we prove that the set of all [Formula: see text]-regular elements of [Formula: see text] forms a semigroup, where [Formula: see text] is a reversible and [Formula: see text]-compatible ring and [Formula: see text] is a countable locally nilpotent ideal of [Formula: see text].

Independent domination polynomial of zero-divisor graphs of commutative rings

Independent domination polynomial of zero-divisor graphs of commutative rings

The extended graph of a module over a commutative ring

In this paper, we associate a simple graph to a module over a commutative ring and call it extended graph, that adjacent condition depends on the properties of annihilator submodules. We determine the structure of modules, when the associated graph is star or complete. Moreover, we look for relationships between the extended graph and the zero-divisor graph of modules.

Strong resolving graph of a zero-divisor graph

Strong resolving graph of a zero-divisor graph

Analysis of Eccentricity-Based Topological Invariants with Zero-Divisor Graphs

Let R = Z ♭ 1 ♭ 2 ♭ 3 × Z q 2 be a commutative ring, where ♭ 1 , ♭ 2 , ♭ 3 are distinct primes, and q is any prime integer. A zero divisor graph J R of ring R is a graph with vertex set consist of zero divisors elements of R and any two vertices a , b are adjacent if and only if a b = 0 . A topological index is a numerical number associated with the graph and may be helpful to correlate the graph with certain of its physical/chemical properties. In this paper, we have computed some eccentricity based topological indices of J R , namely, atom-bond connectivity index ( AB C 5 ), eccentricity-based harmonic index of fourth type ( H 4 J ), geometric-arithmetic eccentricity index ( G A 4 J ), eccentricity-based third Zagreb index, and eccentricity-based first Zagreb index.

Sombor index of zero-divisor graphs of commutative rings

Sombor index of zero-divisor graphs of commutative rings

A new approach to the diameter of zero-divisor graph

A new approach to the diameter of zero-divisor graph