Metric dimension of ideal-intersection graph of the ring Zn

Saha, Laxman; Basak, Mithun; Tiwary, Kalishankar

doi:10.1080/09728600.2021.1962700

Cited by 3 publications

(4 citation statements)

References 8 publications

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“…𝑘 is factored into three or more numbers, then there will be three or more equivalence classes, which means there will exist three points that are not connected to each other. They will have the maximum possible distance of one between each other, which, relying on Lemma 2.4 from [1], is equal to 2. Therefore, the sum…”

Section: Resultsmentioning

confidence: 99%

“…In this article, we consider simple graphs. Laxman Saha, Mithun Basak and Kalishankar Tiwary in [1] show that for a commutative ring 𝑅, ideal-intersection graph is a simple graph with vertices that are non-zero proper ideals of the ring 𝑅 and two vertices (ideals) are adjacent if their intersection is also a non-zero (proper) ideal of 𝑅. Throughout this paper, 𝐺(𝑅) denotes idealintersection graph of the ring 𝑅.…”

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confidence: 99%

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Properties of the ideal-intersection graph of the ring Zn

Utenko

2024

MMJ

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In this paper we study properties of the ideal-intersection graph of the ring Zn. The graph of ideal intersections is a simple graph in which the vertices are non-zero ideals of the ring, and two vertices (ideals) are adjacent if their intersection is also a non-zero ideal of the ring. These graphs can be referred to as the intersection scheme of equivalence classes (See: Laxman Saha, Mithun Basak Kalishankar Tiwary “Metric dimension of ideal-intersection graph of the ring Zn” [1] ).In this article we prove that the triameter of graph is equal to six or less than six. We also describe maximal clique of the ideal-intersection graph of the ring Zn. We prove that the chromatic number of this graph is equal to the sum of the number of elements in the zero equivalence class and the class with the largest number of element. In addition, we demonstrate that eccentricity is equal to 1 or it is equal to 2. And in the end we describe the central vertices in the ideal-intersection graph of the ring Zn.

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Section: Resultsmentioning

confidence: 99%

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confidence: 99%

Properties of the ideal-intersection graph of the ring Zn

Utenko

2024

MMJ

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“…Tomáš related MD to directed and undirected circulant graphs (see [32]). In 2021, MD of ideal-intersection graph of the ring Z n was discussed by Saha et al (see [28]).…”

Section: Introduction and Basic Terminologiesmentioning

confidence: 99%

Fault-Tolerant Partition Resolvability in Mesh Related Networks and Applications

et al. 2022

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Fault-tolerance of a system measures its working capability in the presence of faulty components in the system. The fault-tolerant partition dimension of a network computes the least number of subcomponents of network required to distinctively identify each node in the presence of faults, having promising applications in telecommunication, robot navigation and geographical routing protocols. In this paper, certain triangular mesh networks including, triangular ladder (T l s ), triangular mesh (T s ) , reflection triangular mesh (rl(T s )), tower triangular mesh (T r s ) and reflection tower triangular mesh (rl(T r s )) networks are discussed for their partition and fault-tolerant partition resolvability. In this regard, it is shown that the partition dimension of these networks is 3, whereas their fault-tolerant partition dimension is 4.INDEX TERMS triangular ladder, triangular mesh, metric dimension, partition dimension, fault-tolerant partition dimension.

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“…A ring of integer modulo 𝑛 is denoted by ℤ 𝑛 . Some researchers have also studied a graph with ring of integer modulo 𝑛, as in (Basak et al, 2019), (Mishra & Patra, 2020), and (Ju et al, 2014). In (2021), Any and Hidayah have researched the girth of the total graph of ℤ 𝑛, denoted by 𝑔𝑟(𝑇 𝛤 (ℤ 𝑛 )).…”

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confidence: 99%

Algorithm for Constructing Total Graph of Commutative Ring

Meinawati,

Kurniawan,

Kurdhi

2024

JTAM

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Let R be a commutative ring. The total graph of R, denoted by TΓ(R) is a graph whose vertices are all elements of the ring R and every i,j∈R with i≠j, then i and j vertices are connected by edges if and only if i+j∈Z(R), where Z(R) is the set of zero-divisors in R with 0∈Z(R). Python programming is code that is easy to learn, read, understand, and helpful in explaining problems regarding graphs and algebra. In this paper, we determine an algorithm to construct the total graph of ring Z_n using Python. The research methods in this paper is a literature studies. The results generated by the algorithm can be utilized to observe the characteristic patterns displayed by the graph. Based on the algorithm’s constructed graph pattern, several properties of TΓ(Z_n ) can be inferred. For instance, if n is a prime number, then TΓ(Z_n ) is a disconnected graph. On the other hand, if n is a prime number and n≥3, then TΓ(Z_2n ) and TΓ(Z_4n ) is a connected graph, regular graph, Hamiltonian graph, and has a girth gr(TΓ(〖Z〗_n ))=3. In this paper we creating an algorithm to construct total graphs from commutative rings streamlines the construction process, enhances accessibility and utilization of total graphs, and supports parameter variation exploration and application in problem-solving.

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Metric dimension of ideal-intersection graph of the ring Zn

Abstract: Metric Dimension of a simple connected graph is the minimum number of vertices those are used to identify each vertex of the graph uniquely using distance code. In this paper, we determine metric dimension of ideal-intersection graph for the ring Z n , where n being a positive integer.

Cited by 3 publications

References 8 publications

Properties of the ideal-intersection graph of the ring Zn

Properties of the ideal-intersection graph of the ring Zn

Fault-Tolerant Partition Resolvability in Mesh Related Networks and Applications

Algorithm for Constructing Total Graph of Commutative Ring

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