Nonzero Zero-divisors Research Articles

Some results on the total zero-divisor graph of a commutative ring

PurposeThe purpose of this paper is to characterize a commutative ring R with identity which is not an integral domain such that ZT(R), the total zero-divisor graph of R is connected and to determine the diameter and radius of ZT(R) whenever ZT(R) is connected. Also, the purpose is to generalize some of the known results proved by Duric et al. on the total zero-divisor graph of R.Design/methodology/approachWe use the methods from commutative ring theory on primary decomposition and strong primary decomposition of ideals in commutative rings. The structure of ideals, the primary ideals, the prime ideals, the set of zero-divisors of the finite direct product of commutative rings is used in this paper. The notion of maximal Nagata prime of the zero-ideal of a commutative ring is also used in our discussion.FindingsFor a commutative ring R with identity, ZT(R) is the intersection of the zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R. The zero-divisor graph of R and the total graph of R induced by the set of all non-zero zero-divisors of R are well studied. Hence, we determine necessary and sufficient condition so that ZT(R) agrees with the zero-divisor graph of R (respectively, agrees with the total graph induced by the set of non-zero zero-divisors of R). If Z(R) is an ideal of R, then it is noted that ZT(R) agrees with the zero-divisor graph of R. Hence, we focus on rings R such that Z(R) is not an ideal of R. We are able to characterize R such that ZT(R) is connected under the assumptions that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. With the above assumptions, we are able to determine the domination number of ZT(R).Research limitations/implicationsDuric et al. characterized Artinian rings R such that ZT(R) is connected. In this paper, we extend their result to rings R such that the zero ideal of R admits a strong primary decomposition and Z(R) is not an ideal of R. As an Artinian ring is isomorphic to the direct product of a finite number of Artinian local rings, we try to characterize R such that ZT(R) is connected under the assumption that R is ta finite direct product of rings R1, R2, … Rn with Z(Ri) is an ideal of Ri for each i between 1 to n. Their result on domination number of ZT(R) is also generalized in this paper. We provide several examples to illustrate our results proved.Practical implicationsThe implication of this paper is that the existing result of Duric et al. is applicable to large class of commutative rings thereby yielding more examples. Moreover, the results proved in this paper make us to understand the structure of commutative rings better. It also helps us to learn the interplay between the ring-theoretic properties and the graph-theoretic properties of the graph associated with it.Originality/valueThe results proved in this paper are original and they provide more insight into the structure of total zero-divisor graph of a commutative ring. This paper provides several examples. Not much work done in the area of total zero-divisor graph of a commutative ring. This paper is a contribution to the area of graphs and rings and may inspire other researchers to study the total zero-divisor graph in further detail.

Extended total graph associated to finite commutative rings

UDC 512.5 For a commutative ring R with nonzero identity 1 ≠ 0 , let Z ( R ) denote the set of zero divisors. The total graph of R denoted by T Γ ( R ) is a simple graph in which all elements of R are vertices and any two distinct vertices x and y are adjacent if and only if x + y ∈ Z ( R ) . In this paper, we define an extension of the total graph denoted by T ( Γ e ( R ) ) with vertex set as Z ( R ) , and two distinct vertices x and y are adjacent if and only if x + y ∈ Z * ( R ) , where Z * ( R ) is the set of nonzero zero divisors of R . Our main aim is to characterize the finite commutative rings whose T ( Γ e ( R ) ) has clique numbers 1,2 , and 3 . In addition, we characterize finite commutative nonlocal rings R for which the corresponding graph T ( Γ e ( R ) ) has the clique number 4.

Randić spectrum of the weakly zero-divisor graph of the ring ℤn

In this article, we find the Randić spectrum of the weakly zero-divisor graph of a finite commutative ring R with identity 1 ≠ 0 , denoted as W Γ ( R ) , where R is taken as the ring of integers modulo n . The weakly zero-divisor graph of the ring R is a simple undirected graph with vertices representing non-zero zero-divisors in R . Two vertices, denoted as a and b, are connected if there are elements x in the annihilator of a and y in the annihilator of b such that their product xy equals zero. In particular, we examine the Randić spectrum of W Γ ( Z n ) for specific values of n , which are products of prime numbers and their powers.

Annihilator intersection graph of a lattice

For a lattice [Formula: see text], we associate a graph called the annihilator intersection graph of [Formula: see text], denoted by [Formula: see text] The vertex set of [Formula: see text] is the set of all nonzero zero-divisors of [Formula: see text] and any two distinct vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text]. It has shown that the [Formula: see text] is disconnected if and only if the number of atoms in [Formula: see text] is two. If [Formula: see text] is connected, then we determine the diameter and the girth of [Formula: see text] We characterize all lattices whose annihilator intersection graph is planar. Further, we obtain the clique number and chromatic number of [Formula: see text] when [Formula: see text] is a finite Boolean lattice. We show that the domination number of [Formula: see text] is not exceeding two. Finally, we obtain a condition under which the annihilator intersection graph is identical with the zero-divisor graph and the annihilator ideal graph of lattices.

Graphs from matrices - a survey

Let R be a commutative ring with identity. For a positive integer n ≥ 2 , let M n ( R ) be the set of all n × n matrices over R and M n ( R ) * be the set of all non-zero matrices of M n ( R ) . The zero-divisor graph of M n ( R ) is a simple directed graph with vertex set the non-zero zero-divisors in M n ( R ) and two distinct matrices A and B are adjacent if their product is zero. Given a matrix A ∈ M n ( R ) , Tr(A) is the trace of the matrix A. The trace graph of the matrix ring M n ( R ) , denoted by Γ t ( M n ( R ) ) , is the simple undirected graph with vertex set { A ∈ M n ( R ) * : there exists B ∈ M n ( R ) * such that Tr ( AB ) = 0 } and two distinct vertices A and B are adjacent if and only if Tr ( AB ) = 0. For an ideal I of R, the notion of the ideal based trace graph, denoted by Γ I t ( M n ( R ) ) , is a simple undirected graph with vertex set M n ( R ) ∖ M n ( I ) and two distinct vertices A and B are adjacent if and only if Tr ( AB ) ∈ I . In this survey, we present several results concerning the zero-divisor graph, trace graph and the ideal based trace graph of matrices over R.

On the metric dimension of extended zero divisor graphs

Let R be a commutative ring with unity, the extended zero divisor graph E(R) is defined by considering the non-zero zero divisors Z(R)∗ as the vertices and two distinct vertices x and y of R are adjacent whenever there exist two positive integers m and n such that xmym=0 with xm≠0 and yn≠ 0. In this paper, we investigate metric dimension, fault tolerant metric dimension and local metric dimension of the extended zero divisor graph for the ring of integers modulo m and the ring of Gaussian integers modulo m.

Complemented zero-divisor graphs associated with finite commutative semigroups

Let S be a commutative semigroup with zero. The zero-divisor graph associated with S is the simple graph whose vertex set is given by the nonzero zero-divisors of S where two distinct vertices are adjacent if and only if their product is zero. A graph is called complemented if every vertex has an incident edge which is not the edge of any three cycle. Complemented zero-divisor graphs which are associated with a finite commutative semigroup are described in terms of their clique number. The algebraic properties of the finite semigroups S whose zero-divisor graphs are complemented are described in terms of the clique number of the graph. Infinite families of complemented zero-divisor graphs associated with a finite commutative semigroup are given.

Common zero-divisor graph over a commutative ring

Let [Formula: see text] be a finite commutative ring with non-zero identity and [Formula: see text] be the set of all non-zero zero-divisors of [Formula: see text]. In this paper, for a non-empty subset [Formula: see text] of [Formula: see text], we introduce a new graph, denoted by [Formula: see text], with vertex set [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there is [Formula: see text] such that [Formula: see text]. Then, we investigate some properties of the graph [Formula: see text].

Annihilator graphs derived from group rings

Let RG be the group ring of the group G over a ring R and let Z∗(RG) be the collection of all non-zero zero divisors in a finite group ring RG. And, for x ∈ Z∗(RG), ann(x) = { r ∈ RG | rx = 0}. The annihilator graph of RG denoted as AG(RG), and is defined as the graph whose vertex set is the elements of non-zero zero divisors in RG and two distinct vertices vx and vy are adjacent if and only if ann(x) ∩ ann(y) ≠ 0. In this paper we try to characterize the properties of annihilator graphs in group rings.

The First Zagreb Index of the Zero Divisor Graph for the Ring of Integers Modulo Power of Primes

Let be a simple graph with the set of vertices and edges. The first Zagreb index of a graph is defined as the sum of the degree of each vertex to the power of two. Meanwhile, the zero divisor graph of a ring , denoted by , is defined as a graph with its vertex set contains the nonzero zero divisors in which two distinct vertices and are adjacent if . In this paper, the general formula of the first Zagreb index of the zero divisor graph for the commutative ring of integers modulo , where a prime number and a positive integer is determined. A few examples are given to illustrate the main results.

Wiener index and graph energy of zero divisor graph for commutative rings

Let [Formula: see text] be commutative ring and [Formula: see text] be the set of all non-zero zero divisors of [Formula: see text]. Then [Formula: see text] is said to be the zero divisor graph if and only if [Formula: see text] where [Formula: see text] and [Formula: see text]. Graph energy [Formula: see text] is defined as the sum of the absolute eigenvalues of the adjacency matrix [Formula: see text], then [Formula: see text]. Wiener index [Formula: see text] is defined as the sum of all distance between pairs of vertices [Formula: see text] and [Formula: see text], then [Formula: see text]. In this paper, we compute the graph energy from the adjacency matrix of the zero divisor graph and the Wiener index from the zero divisor graph associated with commutative rings.

The i-extended zero-divisor graphs of idealizations

Let R be a commutative ring with zero-divisors [Formula: see text] and [Formula: see text] be a positive integer. The [Formula: see text]-extended zero-divisor graph of [Formula: see text], denoted by [Formula: see text], is the (simple) graph with vertex set [Formula: see text], the set of nonzero zero-divisors of [Formula: see text], and two distinct nonzero zero-divisors [Formula: see text] and [Formula: see text] are adjacent whenever there exist two positive integers [Formula: see text] such that [Formula: see text] with [Formula: see text] and [Formula: see text]. The [Formula: see text]-extended zero-divisor graph of [Formula: see text] is well studied in 10. In this paper, we characterize the [Formula: see text]-extended zero-divisor graphs of idealizations [Formula: see text] (where [Formula: see text] is an [Formula: see text]-module). Namely, we study in detail the behavior of the filtration [Formula: see text] as well as the relations between its terms. We also characterize the girth and the diameter of [Formula: see text] and we give answers to several interesting and natural questions that arise in this context.

Applications on Topological Indices of Zero-Divisor Graph Associated with Commutative Rings

A topological index is a numeric quantity associated with a chemical structure that attempts to link the chemical structure to various physicochemical properties, chemical reactivity, or biological activity. Let R be a commutative ring with identity, and Z*(R) is the set of all non-zero zero divisors of R. Then, Γ(R) is said to be a zero-divisor graph if and only if a·b=0, where a,b∈V(Γ(R))=Z*(R) and (a,b)∈E(Γ(R)). We define a∼b if a·b=0 or a=b. Then, ∼ is always reflexive and symmetric, but ∼ is usually not transitive. Then, Γ(R) is a symmetric structure measured by the ∼ in commutative rings. Here, we will draw the zero-divisor graph from commutative rings and discuss topological indices for a zero-divisor graph by vertex eccentricity. In this paper, we will compute the total eccentricity index, eccentric connectivity index, connective eccentric index, eccentricity based on the first and second Zagreb indices, Ediz eccentric connectivity index, and augmented eccentric connectivity index for the zero-divisor graph associated with commutative rings. These will help us understand the characteristics of various symmetric physical structures of finite commutative rings.

Universal adjacency spectrum of the looped zero divisor graph for a finite commutative ring with unity

For a finite undirected looped graph [Formula: see text], the universal adjacency matrix [Formula: see text] is a linear combination of the adjacency matrix [Formula: see text], the degree matrix [Formula: see text], the identity matrix [Formula: see text] and the all-ones matrix [Formula: see text], that is [Formula: see text], where [Formula: see text] and [Formula: see text]. For a finite commutative ring [Formula: see text] with unity, the looped zero divisor graph [Formula: see text] is an undirected graph with the set of all nonzero zero divisors of [Formula: see text] as vertices and two vertices (not necessarily distinct) [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we study some structural properties of [Formula: see text] by defining an equivalence relation on its vertex set. Then we obtain the universal adjacency eigenpairs of [Formula: see text], and as a consequence several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree matrix of [Formula: see text] can be obtained in a unified way. Moreover, we get the structural properties and the universal adjacency eigenpairs of the looped zero divisor graph of a reduced ring in a simpler form.

On Connectivity of Zero-Divisor Graphs and Complement Graphs of some Semi-Local Rings

Zero-divisor graphs have been a key area of focus for many researchers. For the semi local ring R of finite cartesian product of finite fields, we consider the zero divisor graph of R denoted by Γ(R) with vertex set as the non-zero zero-divisors of R where two vertices u and v are adjacent if and only if the product of u and v is the additive identity of the Ring R. The objective of this paper is to determine the number of cut vertices and cut edges, vertex connectivity and edge connectivity of the zero divisor graph Γ(R) and complement graph Γ(R).

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.

Twin-free cliques in annihilator graphs of commutative rings

For a connected graph [Formula: see text] a clique [Formula: see text] is twin-free if every pair of elements of [Formula: see text] have distinct closed neighborhoods and the number of elements in a twin-free clique of maximum cardinality is called twin-free clique number of [Formula: see text]. The annihilator graph [Formula: see text] of a commutative and unital ring [Formula: see text] is a graph whose vertices are all non-zero zero-divisors of [Formula: see text] and there is an edge between two distinct vertices [Formula: see text] if and only if [Formula: see text] is properly contained in [Formula: see text]. In this paper, twin-free clique number of [Formula: see text] is computed and as an application the strong metric dimension of [Formula: see text] is characterized. Among other things, for a reduced ring [Formula: see text], the forcing strong metric dimension of [Formula: see text] is given.

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.

On the Aα spectrum of the zero-divisor graphs

Let [Formula: see text] be a commutative ring with nonzero identity and [Formula: see text] be the set of all nonzero zero-divisors of [Formula: see text]. The zero-divisor graph of [Formula: see text], denoted by [Formula: see text], is a simple graph whose vertex set consists of all elements of [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we investigate the [Formula: see text]-spectral of the zero-divisor graph of the ring [Formula: see text]. Specially, we study the [Formula: see text]-spectral of the zero-divisor graph of the ring [Formula: see text] for [Formula: see text], [Formula: see text], [Formula: see text], where [Formula: see text], [Formula: see text] and [Formula: see text] are distinct prime numbers and [Formula: see text]. Moreover, we study the [Formula: see text]-spectral of the zero-divisor graph of the ring [Formula: see text], where [Formula: see text] is a prime number and [Formula: see text].

The Wiener index of the zero-divisor graph of a finite commutative ring with unity

Let R be an arbitrary finite commutative ring with unity. The zero-divisor graph of R, denoted by Γ(R), is a graph with vertex set non-zero zero-divisors of R and two of them are connected by an edge if their product is zero. In this paper, we derive a formula for the Wiener index of the graph Γ(R). In the literature, the Wiener index of the graph Γ(R) is known only for R=Zn, the ring of integers modulo n. As applications of our formula, the Wiener index of Γ(R) is explicitly calculated when (i) R is a reduced ring, (ii) R is the ring of integers modulo n, and (iii) more generally R is the product of ring of integers modulo n. The Wiener index of the zero-divisor graph of the ring of Gaussian integers over Zn is also discussed.