The<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="E1"><mml:mi>k</mml:mi></mml:math>-Zero-Divisor Hypergraph of a Commutative Ring

Eslahchi, Changiz; Rahimi, A. M.

doi:10.1155/2007/50875

Cited by 21 publications

(5 citation statements)

References 9 publications

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“…By extending the idea of Theorem 2.5 of [13], we obtain the following theorem. For the definition of a clique, we refer to [7].…”

Section: Discussionmentioning

confidence: 99%

“…In [7], R is a commutative ring with nonzero identity. A nonzero and nonunit element z 1 of R is called a k-zero-divisor of R if k − 1 distinct nonunit elements z 2 , z 3 , z 4 , .…”

Section: Introductionmentioning

confidence: 99%

“…, z k above must be nonzero elements. Thus, if Z(R, k) is finite, then it is natural to define the k-zero-divisor hypergraph [7] as follows:…”

Section: Introductionmentioning

confidence: 99%

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k-Zero-Divisor and Ideal-Based k-Zero-Divisor Hypergraphs of Some Commutative Rings

Siriwong

Boonklurb

2021

Symmetry

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Let R be a commutative ring with nonzero identity and k≥2 be a fixed integer. The k-zero-divisor hypergraph Hk(R) of R consists of the vertex set Z(R,k), the set of all k-zero-divisors of R, and the hyperedges of the form {a1,a2,a3,⋯,ak}, where a1,a2,a3,⋯,ak are k distinct elements in Z(R,k), which means (i) a1a2a3⋯ak=0 and (ii) the products of all elements of any (k−1) subsets of {a1,a2,a3,⋯,ak} are nonzero. This paper provides two commutative rings so that one of them induces a family of complete k-zero-divisor hypergraphs, while another induces a family of k-partite σ-zero-divisor hypergraphs, which illustrates unbalanced or asymmetric structure. Moreover, the diameter and the minimum length of all cycles or girth of the family of k-partite σ-zero-divisor hypergraphs are determined. In addition to a k-zero-divisor hypergraph, we provide the definition of an ideal-based k-zero-divisor hypergraph and some basic results on these hypergraphs concerning a complete k-partite k-uniform hypergraph, a complete k-uniform hypergraph, and a clique.

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“…By extending the idea of Theorem 2.5 of [13], we obtain the following theorem. For the definition of a clique, we refer to [7].…”

Section: Discussionmentioning

confidence: 99%

“…In [7], R is a commutative ring with nonzero identity. A nonzero and nonunit element z 1 of R is called a k-zero-divisor of R if k − 1 distinct nonunit elements z 2 , z 3 , z 4 , .…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

k-Zero-Divisor and Ideal-Based k-Zero-Divisor Hypergraphs of Some Commutative Rings

Siriwong

Boonklurb

2021

Symmetry

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“…In the literature, there are many papers assigning graphs to ideals of rings, see [10][11][12][13][14]. In [15], Eslahchi and Rahimi have introduced and investigated a graph called the k-zero-divisor hypergraph of a commutative ring. In this view, we define k-annihilating ideal of a commutative ring.…”

Section: Introductionmentioning

confidence: 99%

The -annihilating-ideal hypergraph of commutative ring

Selvakumar

Ramanathan

2019

AKCE International Journal of Graphs and Combinatorics

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The concept of the annihilating-ideal graph of a commutative ring was introduced by Behboodi et. al in 2011. In this paper, we extend this concept to the hypergraph for which we define an algebraic structure called k-annihilating-ideal of a commutative ring which is the vertex set of the hypergraph of such commutative ring. Let R be a commutative ring and k an integer greater than 2 and let A(R, k) be the set of all k-annihilating-ideals of R. The k-annihilating-ideal hypergraph of R, denoted by AG k (R), is a hypergraph with vertex set A(R, k), and for distinct elements I 1 , I 2 ,. .. , I k in A(R, k), the set {I 1 , I 2 ,. .. , I k } is an edge of AG k (R) if and only if k ∏ i=1 I i = (0) and the product of any (k −1) elements of the {I 1 , I 2 ,. .. , I k } is nonzero. In this paper, we provide a necessary and sufficient condition for the completeness of 3-annihilating-ideal hypergraph of a commutative ring. Further, we study the planarity of AG 3 (R) and characterize all commutative ring R whose 3-annihilating-ideal hypergraph AG 3 (R) is planar.

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“…Later in [2], Anderson and Livingston modified this idea when studied the zero divisor graph that have vertices and for , edges if and only if . Many authors studied this notion see for examples [3], [4], [5] and [6] Recently, there are other concepts of zero divisor graph, see for examples [7], [8], [9] and In graph theory " denotes by the eccentricity of a vertex v of a connected graph G which is the number . That means is the distance between v and a vertex furthest from v. The radius of G ,which is denoted by is , while the diameter of G is the maximum eccentricity and it is denoted by .…”

mentioning

confidence: 99%