We prove that if the compressed zero-divisor graph of a finite associative ring contains only one strong vertex then this vertex has a loop. Moreover, more properties of the compressed zero-divisor graph of a finite associative ring are proved.
In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices …
In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring. This paper deals with associative finite rings whose compressed zero-divisor graphs have four vertices. Namely, we find all graphs containing four vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring.
This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDG) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring $R$ …
This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDG) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring $R$ and the associated compressed zero-divisor graph. An undirected graph consisting of a vertex set $ Z(R_E)\backslash\{[0]\} = R_E\backslash\{[0],[1]\}$, where $R_E=\{[x] : x\in R\} $ and $[x]=\{y\in R : \text{ann}(x)=\text{ann}(y)\}$ is called a compressed zero-divisor graph, denoted by $\Gamma_E (R)$. An edge is formed between two vertices $[x]$ and $[y]$ of $Z(R_E)$ if and only if $[x][y]=[xy]=[0]$, that is, iff $xy=0$. For a ring $R$, graph $G$ is said to be realizable as $\Gamma_E (R) $ if $G$ is isomorphic to $\Gamma_E (R)$. We classify the rings based on Mdim of their associated CZDG and obtain the bounds for the Mdim of the compressed zero-divisor graphs. We also study the Mdim of realizable graphs of rings. Moreover, some examples are provided to support our results. Lately, we have discussed the interconnection between Mdim, girth, and diameter of CZDG.
Let $R$ be a commutative ring. In this paper, we introduce and study the compressed essential graph of $R$, $EG_E(R)$. The compressed essential graph of $R$ is a graph whose …
Let $R$ be a commutative ring. In this paper, we introduce and study the compressed essential graph of $R$, $EG_E(R)$. The compressed essential graph of $R$ is a graph whose vertices are equivalence classes of non-zero zero-divisors of $R$ and two distinct vertices $[x]$ and $[y]$ are adjacent if and only if $\ann(xy)$ is an essential ideal of $R$. It is shown if $R$ is reduced, then $EG_E(R)=\Gamma_E(R)$, where $\Gamma_E(R)$ denotes the compressed zero-divisor graph of $R$. Furthermore, for a non-reduced Noetherian ring $R$ with $3<|EG_E(R)|<\infty $, it is shown that $EG_E(R)=\Gamma_E(R)$ if and only if \begin{itemize} \item[(i)] $\Nil(R)=\ann(Z(R))$. \item[(ii)] Every non-zero element of $\Nil(R)$ is irreducible in $Z(R)$. \end{itemize}
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 …
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.
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) …
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.
Let [Formula: see text], [Formula: see text], [Formula: see text] denote the zero-divisor graph, compressed zero-divisor graph and annihilating ideal graph of a commutative ring [Formula: see text], respectively. In …
Let [Formula: see text], [Formula: see text], [Formula: see text] denote the zero-divisor graph, compressed zero-divisor graph and annihilating ideal graph of a commutative ring [Formula: see text], respectively. In this paper, we prove that [Formula: see text] for a semisimple commutative ring [Formula: see text] and represent [Formula: see text] as a generalized join of a finite set of graphs. Further, we study the zero-divisor graph of a semisimple group-ring [Formula: see text] and proved several structural properties of [Formula: see text] and [Formula: see text], where [Formula: see text] is a field with [Formula: see text] elements and [Formula: see text] is a cyclic group with [Formula: see text] elements.
This paper continues the ongoing effort to study the compressed zero-divisor graph over noncommutative rings. The purpose of our paper is to study the diameter of the compressed zero-divisor graph …
This paper continues the ongoing effort to study the compressed zero-divisor graph over noncommutative rings. The purpose of our paper is to study the diameter of the compressed zero-divisor graph of Ore extensions and give a complete characterization of the possible diameters of ?E(R[x; ?,?]), where the base ring R is reversible and also have the (?,?)-compatible property. Also, we give a complete characterization of the diameter of ?E (R[[x;?]]), where R is a reversible, ?-compatible and right Noetherian ring. By some examples, we show that all of the assumptions ?reversiblity?, ?(?,?)-compatiblity? and ?Noetherian? in our main results are crucial.
Let R be a non-commutative ring, and I be an ideal of R. In this paper, we generalize the definition of the zero-divisor graph of R with respect to I, …
Let R be a non-commutative ring, and I be an ideal of R. In this paper, we generalize the definition of the zero-divisor graph of R with respect to I, and define several generalized zero-divisor graphs of R with respect to I. In this paper, we investigate the ring-theoretic properties of R and the graph-theoretic properties of all the generalized zero-divisor graphs. We study some basic properties of these generalized zero-divisor graphs related to the connectedness, the diameter and the girth. We also investigate some properties of these generalized zero-divisor graphs with respect to primal ideals.
Zero-divisor graphs, and more recently, compressed zero-divisor graphs are well represented in the commutative ring literature. In this work, we consider various cut structures, sets of edges or vertices whose …
Zero-divisor graphs, and more recently, compressed zero-divisor graphs are well represented in the commutative ring literature. In this work, we consider various cut structures, sets of edges or vertices whose removal disconnects the graph, in both compressed and non-compressed zero-divisor graphs. In doing so, we connect these graph-theoretic concepts with algebraic notions and provide realization theorems of zero-divisor graphs for commutative rings with identity.
Abstract For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph Γ E (R) with vertex set Z(R E …
Abstract For a commutative ring R with 1 ≠ 0, a compressed zero-divisor graph of a ring R is the undirected graph Γ E (R) with vertex set Z(R E ) \ {[0]} = R E \ {[0], [1]} defined by R E = {[x] : x ∈ R}, where [x] = {y ∈ R : ann(x) = ann(y)} and the two distinct vertices [x] and [y] of Z(RE) are adjacent if and only if [x][y] = [xy] = [0], that is, if and only if xy = 0. In this paper, we study the metric dimension of the compressed zero divisor graph Γ E (R), the relationship of metric dimension between Γ E (R) and Γ(R), classify the rings with same or different metric dimension and obtain the bounds for the metric dimension of Γ E (R). We provide a formula for the number of vertices of the family of graphs given by Γ E (R×𝔽). Further, we discuss the relationship between metric dimension, girth and diameter of Γ E (R).
Recently, a lot of research has been carried out regarding graphs built from algebraic structures, including ring structures. One important example of a graph constructed from a ring is the …
Recently, a lot of research has been carried out regarding graphs built from algebraic structures, including ring structures. One important example of a graph constructed from a ring is the zero divisor graph. For a commutative ring R, the zero divisor graph Γ(R) is defined as a simple graph with vertices that are non-zero zero divisors of R, and two distinct vertices are adjacent if and only if the product of the vertices is equal to zero. In this paper, we investigate the zero divisor graph of the quotient ring ℤp[x]/⟨x5⟩ with prime p. More precisely, we characterize some graph properties, including the order, size, adjacency matrix, degree, distance, diameter, girth, clique number, and chromatic number of Γ(ℤp[x]/⟨x5⟩).
Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The extended zero-divisor graph of $R$ is the undirected (simple) graph $\Gamma'(R)$ …
Let $R$ be a commutative ring with identity, and let $Z(R)$ be the set of zero-divisors of $R$. The extended zero-divisor graph of $R$ is the undirected (simple) graph $\Gamma'(R)$ with the vertex set $Z(R)^*=Z(R)\setminus\{0\}$, and two distinct vertices $x$ and $y$ are adjacent if and only if either $Rx\cap \mathrm{Ann}(y)\neq (0)$ or $Ry\cap \mathrm{Ann}(x)\neq (0)$. It follows that the zero-divisor graph $\Gamma(R)$ is a subgraph of $\Gamma'(R)$. It is proved that $\Gamma'(R)$ is connected with diameter at most two and with girth at most four, if $\Gamma'(R)$ contains a cycle. Moreover, we characterize all rings whose extended zero-divisor graphs are complete or star. Furthermore, we study the affinity between extended zero-divisor graph and zero-divisor graph associated with a commutative ring. For instance, for a non-reduced ring $R$, it is proved that the extended zero-divisor graph and the zero-divisor graph of $R$ are identical to the join of a complete graph and a null graph if and only if $ann_R(Z(R))$ is a prime ideal.
The zero-divisor graph 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 …
The zero-divisor graph 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 give full description of finite rings with regular zero-divisor graphs. We also prove some properties of finite rings such that their zero-divisor graphs satisfy the Dirac condition.
In this note, we introduce the cozero-divisor graph of a commutative ring, which is a dual of the zero-divisor graph ‘in some sense’, and we investigate the properties of this …
In this note, we introduce the cozero-divisor graph of a commutative ring, which is a dual of the zero-divisor graph ‘in some sense’, and we investigate the properties of this graph. Moreover, we study the cozero-divisor graph under some extensions of a commutative ring. We also define and investigate the cozero-divisor graph of a partially ordered set.
The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. Let [Formula: see text] be an associative (not …
The present work aims to exploit the interplay between the algebraic properties of rings and the graph-theoretic structures of their associated graphs. Let [Formula: see text] be an associative (not necessarily commutative) ring. We focus on the domination number of the zero-divisor graph [Formula: see text], the compressed zero-divisor graph [Formula: see text] and the unit graph [Formula: see text]. We find some relations between the domination number of the zero-divisor graph and that of the compressed zero-divisor graph. Moreover, some relations between the domination number of [Formula: see text] and [Formula: see text], as well as the relations between the domination number of [Formula: see text] and [Formula: see text], are studied.
This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring R …
This paper explores the concept of multiset dimensions (Mdim) of compressed zero-divisor graphs (CZDGs) associated with rings. The authors investigate the interplay between the ring-theoretic properties of a ring R and the associated compressed zero-divisor graph. An undirected graph consisting of a vertex set Z(RE)\{[0]}=RE\{[0],[1]}, where RE={[x] :x∈R} and [x]={y∈R : ann(x)=ann(y)} is called a compressed zero-divisor graph, denoted by ΓER. An edge is formed between two vertices [x] and [y] of Z(RE) if and only if [x][y]=[xy]=[0], that is, iff xy=0. For a ring R, graph G is said to be realizable as ΓER if G is isomorphic to ΓER. We classify the rings based on Mdim of their associated CZDGs and obtain the bounds for the Mdim of the compressed zero-divisor graphs. We also study the Mdim of realizable graphs of rings. Moreover, some examples are provided to support our results. Notably, we discuss the interconnection between Mdim, girth, and diameter of CZDGs, elucidating their symmetrical significance.
In this paper, we introduce a new graph whose vertices are the nonzero zero-divisors of commutative ring $R$ and for distincts elements $x$ and $y$ in the set $Z(R)^{\star}$ of …
In this paper, we introduce a new graph whose vertices are the nonzero zero-divisors of commutative ring $R$ and for distincts elements $x$ and $y$ in the set $Z(R)^{\star}$ of the nonzero zero-divisors of $R$, $x$ and $y$ are adjacent if and only if $xy=0$ or $x+y\in Z(R)$. we present some properties and examples of this graph and we study his relation with the zero-divisor graph and with a subgraph of total graph of a commutative ring.
Let $R$ be a commutative ring with unity $1\ne 0$. In this paper, we completely describe the vertex and the edge chromatic number of the compressed zero divisor graph of …
Let $R$ be a commutative ring with unity $1\ne 0$. In this paper, we completely describe the vertex and the edge chromatic number of the compressed zero divisor graph of the ring of integers modulo $n$. We find the clique number of the compressed zero divisor graph $\Gamma_E(\mathbb Z_n)$ of $\mathbb Z_n$ and show that $\Gamma_E(\mathbb Z_n)$ is weakly perfect. We also show that the edge chromatic number of $\Gamma_E(\mathbb Z_n)$ is equal to the largest degree proving that $\Gamma_E(\mathbb Z_n)$ resides in class 1 family of graphs.
Abstract To each commutative ring R we can associate a graph, the zero divisor graph of R, whose vertices are the zero divisors of R, and such that two vertices …
Abstract To each commutative ring R we can associate a graph, the zero divisor graph of R, whose vertices are the zero divisors of R, and such that two vertices are adjacent if their product is zero. Presently, we enumerate the local finite commutative rings whose zero divisor graphs have orientable genus 2. Key Words: Commutative algebraFinite ringsLocal ringsZero divisor graphs2000 Mathematics Subject Classification: Primary 05C25Secondary 13H9913M99 ACKNOWLEDGMENTS This work was supported in part by the National Science Foundation under award number DMS-0552573. Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect those of the National Science Foundation.
To each commutative ring R we can associate a zero divisor graph whose vertices are the zero divisors of R and such that two vertices are adjacent if their product …
To each commutative ring R we can associate a zero divisor graph whose vertices are the zero divisors of R and such that two vertices are adjacent if their product is zero. Detecting isomorphisms among zero divisor graphs can be reduced to the problem of computing the classes of R under a suitable semigroup congruence. Presently, we introduce a strategy for computing this quotient for local rings using knowledge about a generating set for the maximal ideal. As an example, we then compute Γ(R) for several classes of rings; with the results in [4 Bloomfield , N. , Wickham , C. ( 2010 ). Local rings with genus 2 zero divisor graph . Comm. Alg. 38 ( 8 ): 2965 – 2980 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]] these classes include all local rings of order p 4 and p 5 for prime p.
In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices …
In [E. V. Zhuravlev and A. S. Monastyreva, Compressed zero-divisor graphs of finite associative rings, Siberian Math. J. 61(1) (2020) 76–84.], we found the graphs containing at most three vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring. This paper deals with associative finite rings whose compressed zero-divisor graphs have four vertices. Namely, we find all graphs containing four vertices that can be realized as the compressed zero-divisor graphs of some finite associative ring.