On the metric dimension of a zero-divisor graph

Pirzada, S.; Raja, Rameez

doi:10.1080/00927872.2016.1175602

Cited by 47 publications

(27 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The zero divisor graph of commutative rings was first introduced by Beck in [1]. After that, many mathematicians studied such graphs [2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

Maximal Ideal Graph of Commutative Rings

Abdulqadr

2020

eijs

View full text Add to dashboard Cite

In this paper, we introduce and study the notion of the maximal ideal graph of a commutative ring with identity. Let R be a commutative ring with identity. The maximal ideal graph of R, denoted by MG(R), is the undirected graph with vertex set, the set of non-trivial ideals of R, where two vertices I1 and I2 are adjacent if I1 I2 and I1+I2 is a maximal ideal of R. We explore some of the properties and characterizations of the graph.

show abstract

“…The zero divisor graph of commutative rings was first introduced by Beck in [1]. After that, many mathematicians studied such graphs [2][3][4][5].…”

Section: Introductionmentioning

confidence: 99%

Maximal Ideal Graph of Commutative Rings

Abdulqadr

2020

eijs

View full text Add to dashboard Cite

show abstract

“…Recently, there was much work done in computing the metric dimension of graphs associated with algebraic structures. Calculating the metric dimension for the commuting graph of a dihedral group was done in [1], for the zero-divisor graphs of commutative rings in [9,10,12], for the compressed zero-divisor graphs of commutative rings in [13], for total graphs of finite commutative rings in [6], for some graphs of modules in [11] and for annihilator graphs of commutative rings in [15]. Motivated by these papers, we study the metric dimension of another graph associated with a commutative ring.…”

Section: Introductionmentioning

confidence: 99%

On the metric dimension of strongly annihilating-ideal graphs of commutative rings

Soleymanivarniab

Nikandish

Tehranian

2020

Acta Universitatis Sapientiae, Mathematica

View full text Add to dashboard Cite

Let 𝒭 be a commutative ring with identity and 𝒜(𝒭) be the set of ideals with non-zero annihilator. The strongly annihilating-ideal graph of 𝒭 is defined as the graph SAG(𝒭) with the vertex set 𝒜 (𝒭)* = 𝒜 (𝒭) \{0} and two distinct vertices I and J are adjacent if and only if I ∩ Ann(J) ≠ (0) and J ∩ Ann(I) ≠ (0). In this paper, we study the metric dimension of SAG(𝒭) and some metric dimension formulae for strongly annihilating-ideal graphs are given.

show abstract

“…For more on the metric dimension of zero-divisor graphs, graphs determined by the equivalence classes of zero-divisors and ideal based zero-divisor graphs associated with commutative rings see [24,25,26]. In the remaining paper, we discuss the nature of graphs associated with modules and also determine the metric dimensions of these graphs, when M is finite over R. First we have the following definition.…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…The authors in [24,25,26] have discussed various properties of metric dimensions which includes the characterization of all finite rings, examination of two equivalence relations on the vertices of Γ(R), relationship between the resolving set and cut vertices of Γ(R), investigation of metric dimension in Γ(R) when R is a finite product of integral domains, when R is the finite product R 1 × R 2 × · · · × R n , where R 1 , R 2 , . .…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

“…The concept of resolving set, metric representation and metric dimension in terms of locating set, locating code and locating number in zero-divisor graphs associated with commutative rings was introduced in [25] and has been further studied in [24,26]. The authors in [24,25,26] have discussed various properties of metric dimensions which includes the characterization of all finite rings, examination of two equivalence relations on the vertices of Γ(R), relationship between the resolving set and cut vertices of Γ(R), investigation of metric dimension in Γ(R) when R is a finite product of integral domains, when R is the finite product R 1 × R 2 × · · · × R n , where R 1 , R 2 , .…”

Section: Definitions and Preliminariesmentioning

confidence: 99%

See 1 more Smart Citation

On Graphs Associated With Modules Over Commutative Rings

Pirzada¹,

Raja²

2016

Journal of the Korean Mathematical Society

Self Cite

View full text Add to dashboard Cite

Abstract. Let M be an R-module, where R is a commutative ring with identity 1 and let G(V, E) be a graph. In this paper, we study the graphs associated with modules over commutative rings. We associate three simple graphs ann f (Γ(M R )), anns(Γ(M R )) and annt(Γ(M R )) to M called full annihilating, semi-annihilating and star-annihilating graph. When M is finite over R, we investigate metric dimensions in ann f (Γ(M R )), anns(Γ(M R )) and annt(Γ(M R )). We show that M over R is finite if and only if the metric dimension of the graph ann f (Γ(M R )) is finite. We further show that the graphs ann f (Γ(M R )), anns(Γ(M R )) and annt(Γ(M R )) are empty if and only if M is a prime-multiplicationlike R-module. We investigate the case when M is a free R-module, where R is an integral domain and show that the graphs ann f (Γ(M R )), anns(Γ(M R )) and annt(Γ(M R )) are empty if and only if M ∼ = R. Finally, we characterize all the non-simple weakly virtually divisible modules M for which Ann(M ) is a prime ideal and Soc(M ) = 0.

show abstract

On the metric dimension of a zero-divisor graph

Cited by 47 publications

References 14 publications

Maximal Ideal Graph of Commutative Rings

Maximal Ideal Graph of Commutative Rings

On the metric dimension of strongly annihilating-ideal graphs of commutative rings

On Graphs Associated With Modules Over Commutative Rings

Contact Info

Product

Resources

About