Non-coprime graph of a finite group

Mansoori, Farzaneh; Erfanian, Ahmad; Tolue, Behnaz

doi:10.1063/1.4954605

Cited by 18 publications

(9 citation statements)

References 5 publications

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“…Mansoori et al [4] discussed the topic of the noncoprime graph specifically on the properties of the graph, for example, the diameter, girth, domination number, and chromatic numbers as well as independence number, for some finite groups. Simultaneously, the authors also proved that G is planar if and only if G is isomorphic to one of the groups Z 2 , Z 3 , Z 4 , Z 2 × Z 2 , Z 5 , Z 6 or S 3 .…”

Section: Definition 2: [4] Noncoprime Graphmentioning

confidence: 99%

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: 99%

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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“…Some of the graphs visualizing groups are the coprime graph of dihedral groups [1] and quaternion groups [2] on the dihedral group [3]. This coprime graph is introduced by Ma [4], and later the dual of the coprime graph, called non-coprime, that introduced by Mansoori [5], which also studied integer modulo [6] and dihedral group [7]. Some other graphs visualize are the power graph of groups [8] [9], and the intersection graph of groups [10].…”

Section: Introductionmentioning

confidence: 99%

The Harmonic Index and the Gutman Index of Coprime Graph of Integer Group Modulo With Order of Prime Power

Husni

Syafitri

Siboro

et al. 2022

BAREKENG: J. Math. & App.

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In the field of mathematics, there are many branches of study, especially in graph theory, mathematically a graph is a pair of sets, which consists of a non-empty set whose members are called vertices and a set of distinct unordered pairs called edges. One example of a graph from a group is a coprime graph, where a coprime graph is defined as a graph whose vertices are members of a group and two vertices with different x and y are neighbors if only if (|x|,|y|)=1. In this study, the author discusses the Harmonic Index and Gutman Index of Coprime Graph of Integer Group Modulo n. The method used in this research is a literature review and analysis based on patterns formed from several case studies for the value of n. The results obtained from this study are the coprime graph of the group of integers modulo n has the harmonic index of and the Gutman index for where is prime and is a natural number.

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“…The visualization is given some properties like girth, diameter, chromatic number, clique number, and the shape of several groups like the dihedral group, the integer modulo group, or the generalized quaternion group. See [1][2] [3] [4] [5] [6] [7] [8] [9] [10] for more detail.…”

Section: Introductionmentioning

confidence: 99%

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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Non-coprime graph of a finite group

Cited by 18 publications

References 5 publications

Prime Power Noncoprime Graph and Probability for Some Finite Groups

Prime Power Noncoprime Graph and Probability for Some Finite Groups

The Harmonic Index and the Gutman Index of Coprime Graph of Integer Group Modulo With Order of Prime Power

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

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