Cozero-divisor Graph Research Articles

A generalization of the cozero-divisor graph of a commutative ring

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be an ideal of [Formula: see text]. The zero-divisor graph of [Formula: see text] with respect to [Formula: see text], denoted by [Formula: see text], is the graph whose vertices are the set [Formula: see text] with distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. The cozero-divisor graph [Formula: see text] of [Formula: see text] is the graph whose vertices are precisely the non-zero, non-unit 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] and [Formula: see text]. In this paper, we introduced and investigated a new generalization of the cozero-divisor graph [Formula: see text] of [Formula: see text] denoted by [Formula: see text]. In fact, [Formula: see text] is a dual notion of [Formula: see text].

Independent domination polynomial for the cozero divisor graph of the ring of integers modulo n

Independent domination polynomial for the cozero divisor graph of the ring of integers modulo n

On the line graph structure of the cozero-divisor graph of a commutative ring

On the line graph structure of the cozero-divisor graph of a commutative ring

Universal adjacency spectrum of the cozero-divisor graph and its complement on a finite commutative ring with unity

For a finite simple undirected graph [Formula: see text], the universal adjacency matrix [Formula: see text] is a linear combination of the adjacency matrix [Formula: see text], the degree diagonal 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]. The cozero-divisor graph [Formula: see text] of a finite commutative ring [Formula: see text] with unity is a simple undirected graph with the set of all nonzero nonunits of [Formula: see text] as vertices and two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. In this paper, we study structural properties of [Formula: see text] by defining an equivalence relation on its vertex set in terms of principal ideals of the ring [Formula: see text]. Then we obtain the universal adjacency eigenpairs of [Formula: see text] and its complement, and as a consequence one may obtain several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree diagonal matrix of [Formula: see text] and [Formula: see text] in an unified way. Moreover, we get the universal adjacency eigenpairs of the cozero-divisor graph and its complement for a reduced ring and the ring of integers modulo [Formula: see text] in a simpler form.

Signless Laplacian spectrum of the cozero-divisor graph of the commutative ring ℤ𝑛

Abstract Let 𝑅 be a commutative ring with identity 1 ≠ 0 1\neq 0 and let Z ⁢ ( R ) ′ Z(R)^{\prime} be the set of all non-zero and non-unit elements of ring 𝑅. Further, Γ ′ ⁢ ( R ) \Gamma^{\prime}(R) denotes the cozero-divisor graph of 𝑅, is an undirected graph with vertex set Z ⁢ ( R ) ′ Z(R)^{\prime} , and w ∉ z ⁢ R w\notin zR and z ∉ w ⁢ R z\notin wR if and only if two distinct vertices 𝑤 and 𝑧 are adjacent, where q ⁢ R qR is the ideal generated by the element 𝑞 in 𝑅. In this paper, we find the signless Laplacian eigenvalues of the graphs Γ ′ ⁢ ( Z n ) \Gamma^{\prime}(\mathbb{Z}_{n}) for n = p 1 N ⁢ p 2 ⁢ p 3 n=p_{1}^{N}p_{2}p_{3} and p 1 N ⁢ p 2 M ⁢ p 3 p_{1}^{N}p_{2}^{M}p_{3} , where p 1 , p 2 , p 3 p_{1},p_{2},p_{3} are distinct primes and N , M N,M are positive integers. We also show that the cozero-divisor graph Γ ′ ⁢ ( Z p 1 ⁢ p 2 ) \Gamma^{\prime}(\mathbb{Z}_{p_{1}p_{2}}) is a signless Laplacian integral.

Spectrum of the Cozero-Divisor Graph Associated to Ring Zn

Let R be a commutative ring with identity 1≠0 and let Z(R)′ be the set of all non-unit and non-zero elements of ring R. Γ′(R) denotes the cozero-divisor graph of R and is an undirected graph with vertex set Z(R)′, w∉zR, and z∉wR if and only if two distinct vertices w and z are adjacent, where qR is the ideal generated by the element q in R. In this article, we investigate the signless Laplacian eigenvalues of the graphs Γ′(Zn). We also show that the cozero-divisor graph Γ′(Zp1p2) is a signless Laplacian integral.

On the genus of reduced cozero-divisor graph of commutative rings

Let R be a commutative ring with identity and let (x) be the principal ideal generated by \(x\in R.\) Let \(\varOmega (R)^*\) be the set of all nontrivial principal ideals of R. The reduced cozero-divisor graph \(\varGamma _r(R)\) of R is an undirected simple graph with \(\varOmega (R)^*\) as the vertex set and two distinct vertices (x) and (y) in \(\varOmega (R)^*\) are adjacent if and only if \((x)\nsubseteq (y)\) and \((y)\nsubseteq (x)\). In this paper, we characterize all classes of commutative Artinian non-local rings for which the reduced cozero-divisor graph has genus at most one.

On the cozero-divisor graphs associated to rings

Let R be a ring with unity. The cozero-divisor graph of a ring R, denoted by is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of R, and two distinct vertices x and y are adjacent if and only if and In this paper, first we study the Laplacian spectrum of We show that the graph is Laplacian integral. Further, we obtain the Laplacian spectrum of for where and p, q are distinct primes. In order to study the Laplacian spectral radius and algebraic connectivity of we characterized the values of n for which the Laplacian spectral radius is equal to the order of Moreover, the values of n for which the algebraic connectivity and vertex connectivity of coincide are also described. At the final part of this paper, we obtain the Wiener index of for arbitrary n.

Clique and Independence Number of Rough Co-Zero Divisor Graph and Its Applications in Analyzing a Twitter Data

In view of this paper, we have defined Rough Co-zero divisor graph [Formula: see text] of a rough semiring [Formula: see text] for a given approximation space [Formula: see text] where [Formula: see text] is nonempty finite set of objects and [Formula: see text] is an arbitrary equivalence relation on [Formula: see text]. It is proved that the Rough Co-zero divisor graph and the subgraph of Rough Ideal-based rough edge Cayley graph [Formula: see text] are complement to each other. Also the Clique number and Independence number of [Formula: see text] and [Formula: see text] are obtained. The developed concepts are illustrated with suitable examples. A sentiment analysis is also made using [Formula: see text] for a Twitter data.

An ideal-based cozero-divisor graph of a commutative ring

Let $R$ be a commutative ring and let $I$ be an ideal of $R$. In this article, we introduce the cozero-divisor graph $\acute{\Gamma}_I(R)$ of $R$ and explore some of its basic properties. This graph can be regarded as a dual notion of an ideal-based zero-divisor graph.

Sentiment analysis using rough co-zero divisor graph

The article is focused on determining the Domination and Double Domination number of a Rough Co-zero divisor graph 𝐺(𝑍∗(𝐽)) and the Rough ideal based rough edge Cayley graph 𝐺(𝑇∗(𝐽)) of a rough Semiring (𝑇,∆,𝛻) for a specific approximation space 𝐼=(𝑈,𝑅) where 𝑈 is nonempty finite set of objects and 𝑅is an arbitrary equivalence relation on 𝑈. Results are established with suitable example. A detailed Sentiment Analysis is made in a Twitter data using domination and double domination number of 𝐺(𝑍∗(𝐽)) and 𝐺(𝑇∗(𝐽)).

Wiener Index of Rough Co-Zero Divisor Graph of a Rough Semiring

Wiener Index of Rough Co-Zero Divisor Graph of a Rough Semiring

Metric and Strong Metric Dimension in Cozero-Divisor Graphs

Let R be a commutative ring with non-zero identity and $$W^*(R)$$ be the set of all non-zero and non-unit elements of R. The cozero-divisor graph of R, denoted by $$\Gamma ^{\prime }(R)$$ , is a graph with the vertex set $$W^*(R)$$ and two distinct vertices a and b are adjacent if and only if $$a\not \in Rb$$ and $$b\not \in Ra$$ . In this paper, the metric dimension and strong metric dimension of $$\Gamma ^{\prime }(R)$$ are investigated. We compute the exact values of strong metric and metric dimension in cozero-divisor graphs of reduced rings. Moreover, the metric dimension in cozero-divisor graphs of non-reduced rings is discussed.

Sentiment analysis using rough co-zero divisor graph

The article is focused on determining the Domination and Double Domination number of a Rough Co-zero divisor graph 𝐺(𝑍∗(𝐽)) and the Rough ideal based rough edge Cayley graph 𝐺(𝑇∗(𝐽)) of a rough Semiring (𝑇,∆,𝛻) for a specific approximation space 𝐼=(𝑈,𝑅) where 𝑈 is nonempty finite set of objects and 𝑅is an arbitrary equivalence relation on 𝑈. Results are established with suitable example. A detailed Sentiment Analysis is made in a Twitter data using domination and double domination number of 𝐺(𝑍∗(𝐽)) and 𝐺(𝑇∗(𝐽)).

Coloring of cozero-divisor graphs of commutative von Neumann regular rings

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by $$\Gamma ^{\prime }(R)$$ , is a graph with vertices in $$W^*(R)$$ , which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in $$W^*(R)$$ are adjacent if and only if $$a\not \in Rb$$ and $$b\not \in Ra$$ . In this paper, we show that the cozero-divisor graph of a von Neumann regular ring with finite clique number is not only weakly perfect but also perfect. Also, an explicit formula for the clique number is given.

The coloring of the cozero-divisor graph of a commutative ring

Let [Formula: see text] be a commutative ring with unity. The cozero-divisor graph of [Formula: see text] denoted by [Formula: see text] is a graph with the vertex set [Formula: see text], where [Formula: see text] is the set of all nonzero and non-unit 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] and [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote the clique number and the chromatic number of [Formula: see text], respectively. In this paper, we prove that if [Formula: see text] is a finite commutative ring, then [Formula: see text] is perfect. Also, we prove that if [Formula: see text] is a commutative Artinian non-local ring and [Formula: see text] is finite, then [Formula: see text]. For Artinian local ring, we obtain an upper bound for the chromatic number of cozero-divisor graph. Among other results, we prove that if [Formula: see text] is a commutative ring, then [Formula: see text] is a complete bipartite graph if and only if [Formula: see text], where [Formula: see text] and [Formula: see text] are fields. Moreover, we present some results on the complete [Formula: see text]-partite cozero-divisor graphs.

The cozero-divisor graph of a module

Let [Formula: see text] be an associative ring with non-zero identity and [Formula: see text] a non-zero unital left [Formula: see text]-module. The cozero-divisor graph of [Formula: see text], denoted by [Formula: see text], is a graph with vertices [Formula: see text] and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if [Formula: see text] and [Formula: see text]. In this paper, we study some connections between the graph-theoretic properties of [Formula: see text] and algebraic-theoretic properties of [Formula: see text] and [Formula: see text]. Also, we study girth, independence number, clique number and planarity of this graph.

Comparison of graphs associated to a commutative Artinian ring

Let R be a commutative ring with $$1\ne 0$$ and the additive group $$R^+$$ . Several graphs on R have been introduced by many authors, among zero-divisor graph $$\Gamma _1(R)$$ , co-maximal graph $$\Gamma _2(R)$$ , annihilator graph AG(R), total graph $$ T(\Gamma (R))$$ , cozero-divisors graph $$\Gamma _\mathrm{c}(R)$$ , equivalence classes graph $$\Gamma _\mathrm{E}(R)$$ and the Cayley graph $$\mathrm{Cay}(R^+ ,Z^*(R))$$ . Shekarriz et al. (J. Commun. Algebra, 40 (2012) 2798–2807) gave some conditions under which total graph is isomorphic to $$\mathrm{Cay}(R^+ ,Z^*(R))$$ . Badawi (J. Commun. Algebra, 42 (2014) 108–121) showed that when R is a reduced ring, the annihilator graph is identical to the zero-divisor graph if and only if R has exactly two minimal prime ideals. The purpose of this paper is comparison of graphs associated to a commutative Artinian ring. Among the results, we prove that for a commutative finite ring R with $$|\mathrm{Max}(R)|=n \ge 3$$ , $$ \Gamma _1(R) \simeq \Gamma _2(R)$$ if and only if $$R\simeq \mathbb {Z}^n_2$$ ; if and only if $$\Gamma _1(R) \simeq \Gamma _\mathrm{E}(R)$$ . Also the annihilator graph is identical to the cozero-divisor graph if and only if R is a Frobenius ring.

Rings whose cozero-divisor graph has crosscap number at most two

Let [Formula: see text] be a commutative ring with identity. The cozero-divisor graph of [Formula: see text], denoted by [Formula: see text], is a graph whose vertex set is [Formula: see text], the set of all non-zero and non-unit elements of [Formula: see text]. Two distinct vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text] and [Formula: see text]. The Crosscap of a graph [Formula: see text], denoted by [Formula: see text], is the minimum integer [Formula: see text] such that the graph can be embedded in the non-orientable surface [Formula: see text]. The planar graph is called [Formula: see text]-outerplanar if removing all the vertices incident on the outer face yields a [Formula: see text]-outerplanar. The Outerplanarity index of a graph [Formula: see text] is the smallest [Formula: see text] such that [Formula: see text] is [Formula: see text]-outerplanar. In this paper, we characterize the class of rings [Formula: see text] (up to isomorphism) for which [Formula: see text]. Further we characterize all finite rings [Formula: see text] (up to isomorphism) for which [Formula: see text] has an outerplanarity index two.

On the genus of graphs from commutative rings

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertex-set W∗(R), which is the set of all non-zero non-unit elements of R, and two distinct vertices x and y in W∗(R) are adjacent if and only if x∉Ry and y∉Rx, where for z∈R, Rz is the ideal generated by z. In this paper, we determine all isomorphism classes of finite commutative rings R with identity whose Γ′(R) has genus one. Also we characterize all non-local rings for which the reduced cozero-divisor graph Γr(R) is planar.