Strong metric dimension in annihilating-ideal graph of commutative rings

Jalali, Mitra; Nikandish, Reza

doi:10.1007/s00200-022-00574-3

Cited by 3 publications

(2 citation statements)

References 20 publications

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“…The metric dimension and strong metric dimension of the prime ideal sum graph of various classes of rings have been discussed in [1,16]. Various embeddings of the prime ideal sum graphs on surfaces have been investigated in [15].…”

Section: Historical Background and Preliminariesmentioning

confidence: 99%

On the cozero-divisor graphs associated to rings

Praveen

Baloda

Kumar

2022

AKCE International Journal of Graphs and Combinatorics

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Let R be a ring with unity. The idempotent graph G Id (R) of a ring R is an undirected simple graph whose vertices are the set of all the elements of ring R and two vertices x and y are adjacent if and only if x + y is an idempotent element of R. In this paper, we obtain a necessary and sufficient condition on the ring R such that G Id (R) is planar. We prove that G Id (R) cannot be an outerplanar graph. Moreover, we classify all the finite non-local commutative rings R such that G Id (R) is a cograph, split graph and threshold graph, respectively. We conclude that latter two graph classes of G Id (R) are equivalent if and only2010 Mathematics Subject Classification. 05C25.

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Section: Historical Background and Preliminariesmentioning

confidence: 99%

On the cozero-divisor graphs associated to rings

Praveen

Baloda

Kumar

2022

AKCE International Journal of Graphs and Combinatorics

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“…Then various graphs associated with rings have been studied extensively in the literature, viz. inclusion ideal graph [6], total graph [8], zero-divisor graph [9], annihilating-ideal graph [13,19], co-maximal graph [22], etc. Afkhami et al [1] introduced the cozero-divisor graph of a commutative ring, in which they have studied the basic graph-theoretic properties including completeness, girth, clique number, etc.…”

Section: Introductionmentioning

confidence: 99%

Wiener index of the Cozero-divisor graph of a finite commutative ring

Baloda¹,

Praveen²,

Kumar³

et al. 2022

Preprint

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Let R be a ring with unity. The cozero-divisor graph of a ring R, denoted by Γ ′ (R), is an undirected simple graph whose vertices are the set of all non-zero and non-unit elements of R, and two distinct vertices x and y are adjacent if and only if x / ∈ Ry and y / ∈ Rx. In this article, we extend some of the results of [24] to an arbitrary ring. In this connection, we derive a closed-form formula of the Wiener index of the cozero-divisor graph of a finite commutative ring R. As applications, we compute the Wiener index of Γ ′ (R), when either R is the product of ring of integers modulo n or a reduced ring. At the final part of this paper, we provide a SageMath code to compute the Wiener index of the cozero-divisor graph of these class of rings including the ring Zn of integers modulo n. et al. [5] studied the cozero-divisor graph associated to the polynomial ring and the ring of power series. Some of the work associated with the cozero-divisor graphs of rings can be found in [3,4,7,11,23,24,25].Over the recent years, the Wiener index of certain graphs associated with rings have been studied by various authors. The Wiener index of the zero divisor graph of the ring Z n of integers modulo n has been studied in [10]. Recently, Selvakumar et al. [28] calculated the Wiener index of the zero divisor graph for a finite commutative ring with unity. The Wiener index of the cozero-divisor graph of the ring Z n has been obtained in [24]. In order to extend the results of [24] to an arbitrary ring, we study the Wiener index of the cozero-divisor graph of a finite commutative 2020 Mathematics Subject Classification. 05C25, 05C50.

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Computing the (forcing) strong metric dimension in strongly annihilating-ideal graphs

Pazoki

Nikandish

2023

AAECC

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Strong metric dimension in annihilating-ideal graph of commutative rings

Cited by 3 publications

References 20 publications

On the cozero-divisor graphs associated to rings

On the cozero-divisor graphs associated to rings

Wiener index of the Cozero-divisor graph of a finite commutative ring

Computing the (forcing) strong metric dimension in strongly annihilating-ideal graphs

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