On the Total Graph of a Ring and Its Related Graphs: A Survey

Badawi, Ayman

doi:10.1007/978-1-4939-0925-4_3

Cited by 12 publications

(11 citation statements)

References 28 publications

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“…Beck conjectured that χ(G) = ω(G) for the zero divisor graph of R and an arbitrary ring R. But Anderson and Naseer [5] have shown that this is not the case in general, namely they presented an example of a commutative local ring R with 32 elements for which χ(G) > ω(G). After that lots of authors have worked on such graphs and found classes of graphs on which Beck's conjecture holds (see for instance [1,2,3,6,7,8,9,14,15,17]).…”

Section: Introductionmentioning

confidence: 99%

On Beck's Coloring for Measurable Functions

Assari

Rahimi

2021

IJMSI

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We study Beck-like coloring of measurable functions on a measure space Ω taking values in a measurable semigroup ∆. To any measure space Ω and any measurable semigroup ∆, we assign a graph (called a zero-divisor graph) whose vertices are labeled by the classes of measurable functions defined on Ω and having values in ∆, with two vertices f and g adjacent if f • g = 0 a.e.. We show that, if Ω is atomic, then not only the Beck's conjecture holds but also the domination number coincides to the clique number and chromatic number as well. We also determine some other graph properties of such a graph.

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Section: Introductionmentioning

confidence: 99%

On Beck's Coloring for Measurable Functions

Assari

Rahimi

2021

IJMSI

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“…The total graph T C ðRÞ of R is the undirected graph with vertex set R and for distinct x, y 2 R are adjacent if and only if x þ y 2 ZðRÞ: Tamizh Chelvam and Asir [4,5,[15][16][17][18][19] have extensively studied about the total graph of commutative rings. Tamizh Chelvam and Balamurugan [22] studied about the complement of the generalized total graph of Z n : For a complete detail about total graphs, one can refer the survey [6,14].…”

Section: Introductionmentioning

confidence: 99%

Complement of the generalized total graph of fields

Chelvam

Balamurugan

2020

AKCE International Journal of Graphs and Combinatorics

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Let R be a commutative ring and H be a multiplicative prime subset of R. The generalized total graph GT H ðRÞ is the undirected simple graph with vertex set R and two distinct vertices x and y are adjacent if x þ y 2 H: For a field F, H ¼ f0g is the only multiplicative prime subset of F and the corresponding generalized total graph is denoted by GTðFÞ: In this paper, we investigate several graph theoretical properties of GTðFÞ, where GTðFÞ is the complement of the generalized total graph of F. In particular, we characterize all the fields for which GTðFÞ is unicyclic, split, chordal, claw-free, perfect and pancyclic.

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“…The study of algebraic structures, using the properties of graphs, became an exciting research topic in the past twenty years, leading to many fascinating results and questions. In the literature, there are many papers assigning graphs to rings, see [1][2][3][4][5][6][7]. In ring theory, the structure of a ring R is closely tied to ideal's behavior more than elements, and so it is deserving to define a graph with vertex set as ideals instead of elements.…”

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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On the Total Graph of a Ring and Its Related Graphs: A Survey

Cited by 12 publications

References 28 publications

On Beck's Coloring for Measurable Functions

On Beck's Coloring for Measurable Functions

Complement of the generalized total graph of fields

The -annihilating-ideal hypergraph of commutative ring

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