Mathematics and Statistics Vol. 9(6), pp. 859 - 866
DOI: 10.13189/ms.2021.090601
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Relative Coprime Probability and Graph for Some Nonabelian Groups of Small Order and Their Associated Graph Properties

Nurfarah Zulkifli , Nor Muhainiah Mohd Ali ^*
Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310, UTM, Johor Bahru, Johor, Malaysia

ABSTRACT

Let be a finite group. The probability that two selected elements from and from are chosen at random in a way that the greatest common divisor also known as gcd, of the order of and , which is equal to one, is called as the relative coprime probability. Meanwhile, another definition states that the vertices or nodes are the elements of a group and two distinct vertices or nodes are adjacent if and only if their orders are coprime and any of them is in the subgroup of the group and this is called as the relative coprime graph. This research focuses on determining the relative coprime probability and graph for cyclic subgroups of some nonabelian groups of small order and their associated graph properties by referring to the definitions and theorems given by previous researchers. Besides, various results of the relative coprime probability for nonabelian groups of small order are obtained. As for the relative coprime graph, the result shows that the domination number for each group is one whereas the number of edges and the independence number for each group vary. Types of graphs that can be formed are either star graph, planar graph or complete subgraph depending on the order of the subgroup of a group.

KEYWORDS
Relative Coprime Probability, Relative Coprime Graph, Types of Graph, Domination Number, Independence Number

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Nurfarah Zulkifli , Nor Muhainiah Mohd Ali , "Relative Coprime Probability and Graph for Some Nonabelian Groups of Small Order and Their Associated Graph Properties," Mathematics and Statistics, Vol. 9, No. 6, pp. 859 - 866, 2021. DOI: 10.13189/ms.2021.090601.

(b). APA Format:
Nurfarah Zulkifli , Nor Muhainiah Mohd Ali (2021). Relative Coprime Probability and Graph for Some Nonabelian Groups of Small Order and Their Associated Graph Properties. Mathematics and Statistics, 9(6), 859 - 866. DOI: 10.13189/ms.2021.090601.