2013
DOI: 10.1142/s0219498813501132
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Some Results on Cozero-Divisor Graph of a Commutative Ring

Abstract: Let R be a commutative ring with unity. The cozero-divisor graph of R denoted by Γ′(R) is a graph with the vertex set W*(R), where W*(R) is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b are adjacent if and only if a ∉ Rb and b ∉ Ra. In this paper, we show that if Γ′(R) is a forest, then Γ′(R) is a union of isolated vertices or a star. Also, we prove that if Γ′(R) is a forest with at least one edge, then R ≅ ℤ2 ⊕ F, where F is a field. Among other results, it is shown tha… Show more

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Cited by 13 publications
(4 citation statements)
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“…Γ (R) denotes the cozero-divisor graph of R, which is an undirected graph with a vertex set Z(R) , w / ∈ zR, and z / ∈ wR if and only if two distinct vertices w and z are adjacent. For more details on the cozero-divisor graph see, for example, [1,7,8] where further references can be found.…”
Section: Introductionmentioning
confidence: 99%
“…Γ (R) denotes the cozero-divisor graph of R, which is an undirected graph with a vertex set Z(R) , w / ∈ zR, and z / ∈ wR if and only if two distinct vertices w and z are adjacent. For more details on the cozero-divisor graph see, for example, [1,7,8] where further references can be found.…”
Section: Introductionmentioning
confidence: 99%
“…Akbari et al [5], studied the cozero-divisor graph associated with the ring R[x]. Some of the work associated with the cozero-divisor graph on the rings can be found in [3,4,6,7,11,12].…”
Section: Introductionmentioning
confidence: 99%
“…Planar, outerplanar and ring graph cozero-divisor graphs may be found in [4]. Akbari et al gave further results on rings with forest cozero-divisor graphs and diameter of cozero-divisor graphs associated with R[x] and R[[x]] (see [6]). The cozero-divisor graph has also been studied in several other papers (e.g., [5,7,8,12]).…”
Section: Introductionmentioning
confidence: 99%