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

# Prime Power Noncoprime Graph and Probability for Some Finite Groups

Authors
Nurfarah Zulkifli1, *, Nor Muhainiah Mohd Ali1
1Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, UTM, 81310, Johor Bahru, Johor, Malaysia
*Corresponding author. Email: nurfarah3@graduate.utm.my
Corresponding Author
Nurfarah Zulkifli
Available Online 12 December 2022.
DOI
10.2991/978-94-6463-014-5_3How to use a DOI?
Keywords
Prime Power Noncoprime Probability; Prime Power Noncoprime Graph; Dihedral Groups; Quasi-dihedral Groups; Generalized Quaternion Groups
Abstract

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.

Copyright
© 2023 The Author(s)
Open Access
Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

Volume Title
Proceedings of the International Conference on Mathematical Sciences and Statistics 2022 (ICMSS 2022)
Series
Advances in Computer Science Research
Publication Date
12 December 2022
ISBN
10.2991/978-94-6463-014-5_3
ISSN
2352-538X
DOI
10.2991/978-94-6463-014-5_3How to use a DOI?
Copyright
© 2023 The Author(s)
Open Access
Open Access This chapter is licensed under the terms of the Creative Commons Attribution-NonCommercial 4.0 International License (http://creativecommons.org/licenses/by-nc/4.0/), which permits any noncommercial use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

### Cite this article

TY  - CONF
AU  - Nurfarah Zulkifli
AU  - Nor Muhainiah Mohd Ali
PY  - 2022
DA  - 2022/12/12
TI  - Prime Power Noncoprime Graph and Probability for Some Finite Groups
BT  - Proceedings of the International Conference on Mathematical Sciences and Statistics 2022 (ICMSS 2022)
PB  - Atlantis Press
SP  - 11
EP  - 23
SN  - 2352-538X
UR  - https://doi.org/10.2991/978-94-6463-014-5_3
DO  - 10.2991/978-94-6463-014-5_3
ID  - Zulkifli2022
ER  -