The weakly zero-divisor graph of a commutative ring

Nikmehr, M. J.; Azadi, Abdolreza; Nikandish, Reza

doi:10.33044/revuma.1677

Cited by 10 publications

(6 citation statements)

References 10 publications

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“…The following theorem endorses that annihilator graph AG(C P (X)) is indeed a subgraph of the weakly zero divisor graph W Γ(C P (X)). A weakly zero divisor graph W Γ(R) of a commutative ring R is a simple graph with Z(R) * as the set of vertices and two distinct vertices a, b are adjacent if there exists c, d ∈ R such that c ∈ ann(a) \ {0}, d ∈ ann(b) \ {0} and c.d = 0 [9].…”

Section: Annihilator Graph Of C P (X)mentioning

confidence: 99%

Annihilator graph of the ring $C_\mathscr{P}(X)$

Pratip¹,

Acharyya²,

Ray³

2022

Preprint

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In this article, we introduce the annihilator graph of the ring C P (X), denoted by AG(C P (X)) and observe the effect of the underlying Tychonoff space X on various graph properties of AG(C P (X)). AG(C P (X)), in general, lies between the zero divisor graph and weakly zero divisor graph of C P (X) and it is proved that these three graphs coincide if and only if the cardinality of the set of all P-points, X P is ≤ 2. Identifying a suitable induced subgraph of AG(C P (X)), called G(C P (X)), we establish that both AG(C P (X)) and G(C P (X)) share similar graph theoretic properties and have the same values for the parameters, e.g., diameter, eccentricity, girth, radius, chromatic number and clique number. By choosing the ring C P (X) where P is the ideal of all finite subsets of X such that X P is finite, we formulate an algorithm for coloring the vertices of G(C P (X)) and thereby get the chromatic number of AG(C P (X)). This exhibits an instance of coloring infinite graphs by just a finite number of colors. We show that any graph isomorphism ψ : AGas a graph and a graph isomorphism φ : G(C P (X)) → G(C Q (Y )) can be extended to a graph isomorphism ψ : AG(C P (X)) → AG(C Q (Y )) under a mild restriction on the function φ. Finally, we show that atleast for the rings C P (X) with finitely many P-points, so far as the graph properties are concerned, the induced subgraph G(C P (X)) is a good substitute for AG(C P (X).

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Section: Annihilator Graph Of C P (X)mentioning

confidence: 99%

Annihilator graph of the ring $C_\mathscr{P}(X)$

Pratip¹,

Acharyya²,

Ray³

2022

Preprint

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“…In Γ(R), authors mainly focused on its coloring. The weakly zero divisor graph WG(R) is studied by Nikmehr et al in [10]. It is an undirected (simple) graph with a vertex set as Z(R) and for any two distinct vertices u and v, u − v is an edge in WG(R) if and only if there exists p ∈ ann(u) and q ∈ ann (v) such that pq = 0.…”

Section: Introductionmentioning

confidence: 99%

Weakly Zero Divisor Graph of a Lattice

Kulal,

Khairnar,

Masalkar

et al. 2023

Comm. Math. Appl.

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For a lattice L, we associate a graph W ZG(L) called a weakly zero divisor graph of L. The vertex set of W ZG(L) is Z * (L), where Z * (L) = {r ∈ L | r ̸ = 0, ∃ s ̸ = 0 such that r ∧ s = 0} and for any distinct u and v in Z * (L), u − v is an edge in W ZG(L) if and only if there exists p ∈ Ann(u) \ {0} and q ∈ Ann(v) \ {0} such that p ∧ q = 0. In this paper, we determined the diameter, girth, independence number and domination number of W ZG(L). We characterized all lattices whose W ZG(L) is complete bipartite or planar. Also, we find a condition so that W ZG(L) is Eulerian or Hamiltonian. Finally, we study the affinity between the weakly zero divisor graph, the zero divisor graph and the annihilatorideal graph of lattices.

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“…These investigations provide insights into the behavior and properties of these graphs and their relationship to the underlying algebraic structures. Mohammad et al [14] proposed a novel concept known as the weakly zero-divisor graph, denoted by WΓ(R). This graph is defined based on the nonzero zero-divisors of a ring R, where each vertex of the graph corresponds to a nonzero zero-divisor of R. The graph is constructed such that two vertices u and v are adjacent, if and only if there exist elements r ∈ ann(u) * and s ∈ ann(v) * , where ann(u) represents the set of elements in R that annihilate u.…”

Section: Introductionmentioning

confidence: 99%

Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring Zp×Zt×Zs

Rehman,

Alali,

Mir

et al. 2023

Axioms

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Let R be a commutative ring with identity, and Z(R) be the set of zero-divisors of R. The weakly zero-divisor graph of R denoted by WΓ(R) is an undirected (simple) graph with vertex set Z(R)*, and two distinct vertices x and y are adjacent, if and only if there exist r∈ann(x) and s∈ann(y), such that rs=0. Importantly, it is worth noting that WΓ(R) contains the zero-divisor graph Γ(R) as a subgraph. It is known that graph theory applications play crucial roles in different areas one of which is chemical graph theory that deals with the applications of graph theory to solve molecular problems. Analyzing Zagreb indices in chemical graph theory provides numerical descriptors for molecular structures, aiding in property prediction and drug design. These indices find applications in QSAR modeling and chemical informatics, contributing to efficient compound screening and optimization. They are essential tools for advancing pharmaceutical and material science research. This research article focuses on the basic properties of the weakly zero-divisor graph of the ring Zp×Zt×Zs, denoted by WΓ(Zp×Zt×Zs), where p, t, and s are prime numbers that may not necessarily be distinct and greater than 2. Moreover, this study includes the examination of various indices and coindices such as the first and second Zagreb indices and coindices, as well as the first and second multiplicative Zagreb indices and coindices of WΓ(Zp×Zt×Zs).

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The weakly zero-divisor graph of a commutative ring

Cited by 10 publications

References 10 publications

Annihilator graph of the ring $C_\mathscr{P}(X)$

Annihilator graph of the ring $C_\mathscr{P}(X)$

Weakly Zero Divisor Graph of a Lattice

Analysis of the Zagreb Indices over the Weakly Zero-Divisor Graph of the Ring Zp×Zt×Zs

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