The zero-divisor graph of a commutative semigroup

DeMeyer, Frank; McKenzie, Thomas C.; Schneider, Kim

doi:10.1007/s002330010128

Cited by 193 publications

(112 citation statements)

References 1 publication

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“…But from Theorem 2.2, we have (1,4), (1,6), (1,8), (2,2), (2,4), (2,6), (2,8), (3,2), (3,4), (3,6), (3,8), (4,2), (4,4), (4,6), (4,8) 1,2), (1,4), (1,5), (1,6), (1,8), (2,2), (2,4), (2,5), (2,6), (2,8), (3,2), (3,4), (3,5), (3,6), (3,…”

Section: Theorem 22mentioning

confidence: 97%

“…In this case V = Z(R 1 )* = {(1,0), (2,0), (1,3), (1,6), (2,3), (2,6) The closed neighborhoods of the vertices are (1,3), (1,6), (2,3), (2,6) …”

Section: Properties Of Adjacency Matrix Mmentioning

confidence: 99%

“…Livinsgston [1] redefined the concept of zero-divisor graph in 1999. F. R. DeMeyer, T. Mckenzie and K. Schneider [3] extended the concept of zero-divisor graph for commutative semi-group in 2002. The notion of zerodivisor graph had been extended for non-commutative rings by S. P. Redmond [9] in 2002.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Adjacency Matrix and Neighborhood Associated with Zero-divisor Graph for Direct Product of Finite Commutative Rings

Patra¹,

Baruah²

2013

IJCATR

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Abstract:The main purpose of this paper is to study the zero-divisor graph for direct product of finite commutative rings. In our present investigation we discuss the zero-divisor graphs for the following direct products: direct product of the ring of integers under addition and multiplication modulo p and the ring of integers under addition and multiplication modulo p 2 for a prime number p, direct product of the ring of integers under addition and multiplication modulo p and the ring of integers under addition and multiplication modulo 2p for an odd prime number p and direct product of the ring of integers under addition and multiplication modulo p and the ring of integers under addition and multiplication modulo p 2 -2 for that odd prime p for which p 2 -2 is a prime number. The aim of this paper is to give some new ideas about the neighborhood, the neighborhood number and the adjacency matrix corresponding to zero-divisor graphs for the above mentioned direct products. Finally, we prove some results of annihilators on zerodivisor graph for direct product of A and B for any two commutative rings A and B with unity

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Section: Theorem 22mentioning

confidence: 97%

“…In this case V = Z(R 1 )* = {(1,0), (2,0), (1,3), (1,6), (2,3), (2,6) The closed neighborhoods of the vertices are (1,3), (1,6), (2,3), (2,6) …”

Section: Properties Of Adjacency Matrix Mmentioning

confidence: 99%

See 1 more Smart Citation

On the Adjacency Matrix and Neighborhood Associated with Zero-divisor Graph for Direct Product of Finite Commutative Rings

Patra¹,

Baruah²

2013

IJCATR

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“…For a given ideal I of a commutative ring R, he defined an undirected graph Γ I (R) with vertices {x ∈ R\I : xy ∈ I f or some y ∈ R\I}, where distinct vertices x and y are adjacent if and only if xy ∈ I. The zero-divisor graph of various algebraic structures has been studied by several authors [ [4], [5], [7] and [11]]. …”

Section: Introductionmentioning

confidence: 99%

Poset Properties Determined by the Ideal - Based Zero-divisor Graph

Porselvi¹,

Elavarasan²

2014

Kyungpook mathematical journal

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“…In 2002, DeMeyer, Mckenzie and Schneider began the study of zero-divisor graph of a commutative semigroup with 0 in [6]. Since then, much work has been done and this becomes a lively branch in semigroup theory and graph theory, see e.g.…”

Section: Introductionmentioning

confidence: 99%

A Construction of Commutative Nilpotent Semigroups

Liu¹,

Wu²,

Ye³

2013

Bulletin of the Korean Mathematical Society

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Abstract. In this paper, we construct nilpotent semigroups S such that S n = {0}, S n−1 = {0} and Γ(S) is a refinement of the star graph K 1,n−3 with center c together with finitely many or infinitely many end vertices adjacent to c, for each finite positive integer n ≥ 5. We also give counting formulae to calculate the number of the mutually non-isomorphic nilpotent semigroups when n = 5, 6 and in finite cases.

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The zero-divisor graph of a commutative semigroup

Cited by 193 publications

References 1 publication

On the Adjacency Matrix and Neighborhood Associated with Zero-divisor Graph for Direct Product of Finite Commutative Rings

On the Adjacency Matrix and Neighborhood Associated with Zero-divisor Graph for Direct Product of Finite Commutative Rings

Poset Properties Determined by the Ideal - Based Zero-divisor Graph

A Construction of Commutative Nilpotent Semigroups

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