On the Line Graph for Zero-Divisors of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mi>C</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>X</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>

Al-Afifi, Ghada; Osba, Emad Abu

doi:10.1155/2013/756179

Cited by 2 publications

(1 citation statement)

References 6 publications

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“…We would like to mention in this context as far as we dig into literature that there are only four papers on graphs having their vertices lying in C(X). See the articles [4], [6], [7] and [8] in this context. In the technical section 2 of this paper we introduce several well-known parameters related to the graph Γ P (X).…”

Section: Introductionmentioning

confidence: 99%

Zero-divisor graph of the rings $C_\mathscr{P}(X)$ and $C^\mathscr{P}_\infty(X)$

Acharyya¹,

Ray²,

Pratip³

2021

Preprint

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In this article we introduce the zero-divisor graphs ΓP (X) and Γ P ∞ (X) of the two rings CP (X) and C P ∞ (X); here P is an ideal of closed sets in X and CP (X) is the aggregate of those functions in C(X), whose support lie on P . C P ∞ (X) is the P analogue of the ring C∞(X). We find out conditions on the topology on X, under-which ΓP (X) (respectively, Γ P ∞ (X)) becomes triangulated/ hypertriangulated. We realize that ΓP (X) (respectively, Γ P ∞ (X)) is a complemented graph if and only if the space of minimal prime ideals in CP (X) (respectively Γ P ∞ (X)) is compact. This places a special case of this result with the choice P ≡ the ideals of closed sets in X, obtained by Azarpanah and Motamedi in [6] on a wider setting. We also give an example of a nonlocally finite graph having finite chromatic number. Finally it is established with some special choices of the ideals P and Q on X and Y respectively that the rings CP (X) and CQ (Y ) are isomorphic if and only if ΓP (X) and ΓQ (Y ) are isomorphic.

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Section: Introductionmentioning

confidence: 99%

Zero-divisor graph of the rings $C_\mathscr{P}(X)$ and $C^\mathscr{P}_\infty(X)$

Acharyya¹,

Ray²,

Pratip³

2021

Preprint

View full text Add to dashboard Cite

show abstract

Zero-divisor graph of the rings Cp(x) and Cp∞ (x)

Acharyya

Ray

Pratip

2022

Filomat

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In this article we introduce the zero-divisor graphs ?P(X) and ?P? (X) of the two rings CP(X) and CP? (X); here P is an ideal of closed sets in X and CP(X) is the aggregate of those functions in C(X), whose support lie on P. CP? (X) is the P analogue of the ring C?(X). We determine when the weakly zero-divisor graph W?P(X) of CP(X) coincides with ?P(X). We find out conditions on the topology on X, under-which ?P(X) (respectively, ?P? (X)) becomes triangulated/ hypertriangulated. We realize that ?P(X) (respectively, ?P? (X)) is a complemented graph if and only if the space of minimal prime ideals in CP(X) (respectively ?P? (X)) is compact. This places a special case of this result with the choice P ? the ideals of closed sets in X, obtained by Azarpanah and Motamedi in [8] on a wider setting. We also give an example of a non-locally finite graph having finite chromatic number. Finally it is established with some special choices of the ideals P and Q on X and Y respectively that the rings CP(X) and CQ(Y) are isomorphic if and only if ?P(X) and ?Q(Y) are isomorphic.

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On the Line Graph for Zero-Divisors of C(X)

Cited by 2 publications

References 6 publications

Zero-divisor graph of the rings $C_\mathscr{P}(X)$ and $C^\mathscr{P}_\infty(X)$

Zero-divisor graph of the rings $C_\mathscr{P}(X)$ and $C^\mathscr{P}_\infty(X)$

Zero-divisor graph of the rings Cp(x) and Cp∞ (x)

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