Some results of non-coprime graph of the dihedral group D2n for n a prime power

Misuki, Wahyu Ulyafandhie; Wardhana, I Gede Adhitya Wisnu; Switrayni, Ni Wayan; Irwansyah,

doi:10.1063/5.0042587

Cited by 11 publications

(7 citation statements)

References 2 publications

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“…In their paper, they also proved that the bounds of the domination number, independence number, and independent domination number are sharp. In addition to the research on noncoprime graph, other interesting results also have been obtained by Misuki et al [6] and in their paper, a different scope of group has been used, which is the dihedral group, D n , in which n is a prime power. The results were separated into two cases.…”

Section: Definition 2: [4] Noncoprime Graphmentioning

confidence: 96%

Prime Power Noncoprime Graph and Probability for Some Finite Groups

Zulkifli¹,

Ali²

2022

Proceedings of the International Conference on Mathematical Sciences and Statistics 2022 (ICMSS 2022)

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The study of coprime probabilities and graphs have its own uniqueness that produces a particular pattern according to its variabilities. Some obvious results can be seen from previous research where the domination number will always be equal to one and the types of graphs that can be formed are either star, planar or r-partite graph depending on certain cases. For the probability, the results vary according to the groups and certain cases need to be considered. The noncoprime graph has been introduced and it is defined as a graph associated to the group G with vertex set G\{e} such that it is possible that two separate vertices are adjacent when the orders are relatively noncoprime. However, in probability theory, the study of noncoprime probability of a group has not been introduced yet. Hence, a thorough study has been conducted where the goal of this research is to introduce a newly defined graph and probability which are the prime power noncoprime graph and prime power noncoprime probability of a group. The focus of this approach is that the greatest common divisor of the order of x and y, where x and y are in G, is equal to a power of prime number. In this paper, the scope of the group is mainly focused on some dihedral groups, quasi-dihedral groups, and some generalized quaternion groups. Some invariants, which are the diameter, girth, clique number, chromatic number, domination number, and independence number of prime power noncoprime graph are found. Additionally, the generalization of the prime power noncoprime probability are also obtained.

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Section: Definition 2: [4] Noncoprime Graphmentioning

confidence: 96%

Prime Power Noncoprime Graph and Probability for Some Finite Groups

Zulkifli¹,

Ali²

2022

Proceedings of the International Conference on Mathematical Sciences and Statistics 2022 (ICMSS 2022)

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“…𝛽 0 = {𝑒, 𝑏} 𝛽 1 = {𝑒, 𝑎𝑏} 𝛽 2 = {𝑒, 𝑎 2 𝑏} 𝛽 3 = {𝑒, 𝑎 3 𝑏} 𝛽 4 = {𝑒, 𝑎 4 𝑏} 𝛽 5 = {𝑒, 𝑎 5 𝑏} 𝛽 6 = {𝑒, 𝑎 6 𝑏} 𝛽 7 = {𝑒, 𝑎 7 𝑏}…”

Section: Reflexion Subgroups (𝜷)unclassified

“…𝛾 10 = {𝑒, 𝑎 2 , 𝑎 4 , 𝑎 6 , 𝑏, 𝑎 2 𝑏, 𝑎 4 𝑏, 𝑎 6 𝑏} 𝛾 11 = {𝑒, 𝑎 2 , 𝑎 4 , 𝑎 6 , 𝑎𝑏, 𝑎 3 𝑏, 𝑎 5 𝑏, 𝑎 7 𝑏}…”

Section: Mixed Subgroups (𝜸)unclassified

The Intersection Graph Representation of a Dihedral Group With Prime Order and Its Numerical Invariants

Ramdani

Wardhana

Awanis

2022

BAREKENG: J. Math. & App.

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One of the concepts in mathematics that developing rapidly today is Graph Theory. The development of Graph Theory has been combined with Group Theory, that is by representing a group in a graph. The intersection graph from group , noted by , is a graph whose vertices are all non-trivial subgroups of group and two distinct vertices are adjacent in if and only if . In this research the intersection graph of a Dihedral group, we looking for the shapes and numerical invariants. The results obtained are if for , then has a subgraphs and subgraphs , the girth of the graph is 3, radius and diameter of the graph in a row is 2 and 3, and the chromatic number of the graph is

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“…For a finite group G, we can represent G in some simple graphs such as the coprime graph, the non-coprime graph, the power graph, the intersection graph, the commuting graph, and others. See [1] [2] [4] [11] [12] for details. In 2014, Ma et al [1] introduced the coprime graph of a finite group G, where the vertex set of the graph is G and two distinct vertices x and y are adjacent if and only if the order of x and the order of y are relative primes [1].…”

Section: Introductionmentioning

confidence: 99%

Numerical Invariants of Coprime Graph of A Generalized Quaternion Group

Nurhabibah

Wardhana

Switrayni

2023

JIMS

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The coprime graph of a finite group was defined by Ma, denoted by ΓG, is a graph with vertices that are all elements of group G and two distinct vertices x and y are adjacent if and only if (|x|, |y|) = 1. In this study, we discuss numerical invariants of a generalized quaternion group. The numerical invariant is a property of a graph in numerical value and that value is always the same on an isomorphic graph. This research is fundamental research and analysis based on patterns in some examples. Some results of this research are the independence number of ΓQ4n is 4n − 1 or 3n and its complement metric dimension is 4n − 2 for each n ≥ 2.

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Some results of non-coprime graph of the dihedral group D2n for n a prime power

Cited by 11 publications

References 2 publications

Prime Power Noncoprime Graph and Probability for Some Finite Groups

Prime Power Noncoprime Graph and Probability for Some Finite Groups

The Intersection Graph Representation of a Dihedral Group With Prime Order and Its Numerical Invariants

Numerical Invariants of Coprime Graph of A Generalized Quaternion Group

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