Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets

Morang’a Daniel Orina; Nyaga Lewis Namu; Gikunju David Muriuki

doi:doi:10.11648/j.engmath.20240801.12

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Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets

Morang’a Daniel Orina^*

, Nyaga Lewis Namu

, Gikunju David Muriuki

Published in Engineering Mathematics (Volume 8, Issue 1)

Received: 17 September 2024 Accepted: 6 October 2024 Published: 31 October 2024

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Abstract

The combinatorial properties and invariants which includes transitivity, primitivity, ranks and subdegrees of direct product of Alternating group and Cyclic group acting on Cartesian product of two set have been extensively studied. However, the construction of suborbital graphs for this group action remains largely unexplored. As a result, this research paper addresses this gap by constructing suborbital graphs involving direct product of the Alternating group and Cyclic group acting on the Cartesian product of two sets where their respective properties such as connectivity, self-pairedness, girth and vertex degree are analyzed in detail. First, the result shows that all suborbits are self − paired implies that the vertex sets are undirected to each other. Secondly, the constructed suborbital graphs are classified into three parts for any value of n ≥ 3. In part A, it is proven that the constructed suborbital graphs Γ₁, Γ₂, Γ₃,..., Γ_(n−1) are undirected, regular of degree (n−1), disconnected with n−connected components and has a girth of 3. In part B, the constructed suborbital graph Γ_(n−1)+1 is found to be undirected, regular of degree (n − 1), disconnected with n − connected components and has a girth of 3. Lastly in part C, the graphs Γ_(n−1)+2, Γ_(n−1)+3, Γ_(n−1)+4,..., Γ_(n−1)+n are found to be undirected, regular of degree (n − 1)², disconnected with n²− connected components and has a girth of 3.

Published in	Engineering Mathematics (Volume 8, Issue 1)
DOI	10.11648/j.engmath.20240801.12
Page(s)	8-17
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Suborbits, Suborbital Graphs, Direct Product, Cartesian Product, Alternating Group, Cyclic Group

References

[1]	Hungerford, Thomas. W. (1974). Groups. Algebra, First Edition. New York: Springer; 23-69. https://doi.org/10.1007/978-1-4612-6101-8_2
[2]	Kurzweil, H. and Stellmacher, B. (2004). Permutation Groups. The Theory of Finite Groups (pp. 77-97). New York: Springer New York. https://doi.org/10.1007/0-387-21768-1_4
[3]	Gallian, J. A. (2021). Contemporary Abstract Algebra (10th ed., pp. 83-88). Cengage Learning. https://doi.org/10.1201/9781003142331
[4]	Gikunju, M. D., Nyaga, N. L., Rimberia, K. J. (2017). Ranks, SubdegreeandSuborbitalGraphofDirectProduct of Symmetric Group Acting on the Cartesian Product of Three Sets. Pure and Applied Mathematics Journal, 6(1), 1-4. https://doi.org/10.11648/j.pamj.20170601.11
[5]	Denton, T. (2022). Introduction to Algebraic Structures. Fields Institute of York University in Toronto LibreTexts, 54-55.
[6]	Gachimu, R., Kamuti, I., Nyaga, L., Rimberia, J., Kamaku, P. (2016). Properties and Invariants Associated with the Action of the Alternating Group on Unordered Subsets. International Journal of Pure and Applied Mathematics, 106(1), 333-346. https://doi.org/10.12732/ijpam.v106i1.27
[7]	Sims, C. C. (1967). Graphs and Finite Permutation Groups. Mathematische Zeitschrift, 95(1), 76-86. https://doi.org/10.1007/bf01117534
[8]	Wielandt, H. (1964). Multiply Transitive Groups. Finite Permutation Groups, 19-43. https://doi.org/10.1016/b978-0-12-749650-4.50007-2
[9]	Cameron, P. J. (1973). Primitive Groups with most Suborbits Doubly Transitive. Geometriae Dedicata, (1), 434-446. https://doi.org/10.1007/bf00184629
[10]	Rosen, K. H. (2012). Discrete Mathematics and Its Applications (7th ed.). McGraw-Hill Education.
[11]	West, D. B. (2001). Introduction to Graph Theory (2nd ed.). Prentice Hall.
[12]	Chartrand, G. and Zhang, P. (2019). Introduction to Graphs. Chromatic Graph Theory, 27-52. https://doi.org/10.1201/9780429438868-2
[13]	Gross, J. L., and Yellen, J. (2005). Graph Theory and Its Applications. Chapman and Hall/CRC. https://doi.org/10.1201/9781420057140
[14]	Nyaga, L. N., Kamuti, I. N., Mwathi, C. W., Akanga, J. R. (2011). Ranks and Subdegrees of the Symmetric Group SnActing on Unordered r-element Subsets. International Journal of Pure and Applied Mathematics, 3(2), 147-163.
[15]	Morang’a, D. O., Nyaga, L. N., Gikunju, D. M. (2024). Combinatorial Properties, Invariants and Structures Associated with the Direct Product of Alternating and Cyclic Groups Acting on the Cartesian Product of Two Sets. American Journal of Applied Mathematics. 12(5), 167-174. https://doi.org/10.11648/j.ajam.20241205.16

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APA Style

Orina, M. D., Namu, N. L., Muriuki, G. D. (2024). Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets. Engineering Mathematics, 8(1), 8-17. https://doi.org/10.11648/j.engmath.20240801.12

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ACS Style

Orina, M. D.; Namu, N. L.; Muriuki, G. D. Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets. Eng. Math. 2024, 8(1), 8-17. doi: 10.11648/j.engmath.20240801.12

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AMA Style

Orina MD, Namu NL, Muriuki GD. Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets. Eng Math. 2024;8(1):8-17. doi: 10.11648/j.engmath.20240801.12

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@article{10.11648/j.engmath.20240801.12,
  author = {Morang’a Daniel Orina and Nyaga Lewis Namu and Gikunju David Muriuki},
  title = {Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets},
  journal = {Engineering Mathematics},
  volume = {8},
  number = {1},
  pages = {8-17},
  doi = {10.11648/j.engmath.20240801.12},
  url = {https://doi.org/10.11648/j.engmath.20240801.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20240801.12},
  abstract = {The combinatorial properties and invariants which includes transitivity, primitivity, ranks and subdegrees of direct product of Alternating group and Cyclic group acting on Cartesian product of two set have been extensively studied. However, the construction of suborbital graphs for this group action remains largely unexplored. As a result, this research paper addresses this gap by constructing suborbital graphs involving direct product of the Alternating group and Cyclic group acting on the Cartesian product of two sets where their respective properties such as connectivity, self-pairedness, girth and vertex degree are analyzed in detail. First, the result shows that all suborbits are self − paired implies that the vertex sets are undirected to each other. Secondly, the constructed suborbital graphs are classified into three parts for any value of n ≥ 3. In part A, it is proven that the constructed suborbital graphs Γ1, Γ2, Γ3,..., Γ(n−1) are undirected, regular of degree (n−1), disconnected with n−connected components and has a girth of 3. In part B, the constructed suborbital graph Γ(n−1)+1 is found to be undirected, regular of degree (n − 1), disconnected with n − connected components and has a girth of 3. Lastly in part C, the graphs Γ(n−1)+2, Γ(n−1)+3, Γ(n−1)+4,..., Γ(n−1)+n are found to be undirected, regular of degree (n − 1)2, disconnected with n2− connected components and has a girth of 3. },
 year = {2024}
}

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TY  - JOUR
T1  - Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets
AU  - Morang’a Daniel Orina
AU  - Nyaga Lewis Namu
AU  - Gikunju David Muriuki
Y1  - 2024/10/31
PY  - 2024
N1  - https://doi.org/10.11648/j.engmath.20240801.12
DO  - 10.11648/j.engmath.20240801.12
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 8
EP  - 17
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20240801.12
AB  - The combinatorial properties and invariants which includes transitivity, primitivity, ranks and subdegrees of direct product of Alternating group and Cyclic group acting on Cartesian product of two set have been extensively studied. However, the construction of suborbital graphs for this group action remains largely unexplored. As a result, this research paper addresses this gap by constructing suborbital graphs involving direct product of the Alternating group and Cyclic group acting on the Cartesian product of two sets where their respective properties such as connectivity, self-pairedness, girth and vertex degree are analyzed in detail. First, the result shows that all suborbits are self − paired implies that the vertex sets are undirected to each other. Secondly, the constructed suborbital graphs are classified into three parts for any value of n ≥ 3. In part A, it is proven that the constructed suborbital graphs Γ1, Γ2, Γ3,..., Γ(n−1) are undirected, regular of degree (n−1), disconnected with n−connected components and has a girth of 3. In part B, the constructed suborbital graph Γ(n−1)+1 is found to be undirected, regular of degree (n − 1), disconnected with n − connected components and has a girth of 3. Lastly in part C, the graphs Γ(n−1)+2, Γ(n−1)+3, Γ(n−1)+4,..., Γ(n−1)+n are found to be undirected, regular of degree (n − 1)2, disconnected with n2− connected components and has a girth of 3. 
VL  - 8
IS  - 1
ER  -

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Author Information

Morang’a Daniel Orina

Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0005-9505-7169
Nyaga Lewis Namu

Department of Pure and Applied Mathematics, Jomo Kenyatta University of Agriculture and Technology, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0000-0004-9187
Gikunju David Muriuki

Department of Mathematical Sciences, The Co-operative University of Kenya, Nairobi, Kenya

Contact Email

http://orcid.org/0009-0001-0285-6692

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APA Style

Orina, M. D., Namu, N. L., Muriuki, G. D. (2024). Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets. Engineering Mathematics, 8(1), 8-17. https://doi.org/10.11648/j.engmath.20240801.12

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ACS Style

Orina, M. D.; Namu, N. L.; Muriuki, G. D. Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets. Eng. Math. 2024, 8(1), 8-17. doi: 10.11648/j.engmath.20240801.12

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AMA Style

Orina MD, Namu NL, Muriuki GD. Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets. Eng Math. 2024;8(1):8-17. doi: 10.11648/j.engmath.20240801.12

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@article{10.11648/j.engmath.20240801.12,
  author = {Morang’a Daniel Orina and Nyaga Lewis Namu and Gikunju David Muriuki},
  title = {Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets},
  journal = {Engineering Mathematics},
  volume = {8},
  number = {1},
  pages = {8-17},
  doi = {10.11648/j.engmath.20240801.12},
  url = {https://doi.org/10.11648/j.engmath.20240801.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.engmath.20240801.12},
  abstract = {The combinatorial properties and invariants which includes transitivity, primitivity, ranks and subdegrees of direct product of Alternating group and Cyclic group acting on Cartesian product of two set have been extensively studied. However, the construction of suborbital graphs for this group action remains largely unexplored. As a result, this research paper addresses this gap by constructing suborbital graphs involving direct product of the Alternating group and Cyclic group acting on the Cartesian product of two sets where their respective properties such as connectivity, self-pairedness, girth and vertex degree are analyzed in detail. First, the result shows that all suborbits are self − paired implies that the vertex sets are undirected to each other. Secondly, the constructed suborbital graphs are classified into three parts for any value of n ≥ 3. In part A, it is proven that the constructed suborbital graphs Γ1, Γ2, Γ3,..., Γ(n−1) are undirected, regular of degree (n−1), disconnected with n−connected components and has a girth of 3. In part B, the constructed suborbital graph Γ(n−1)+1 is found to be undirected, regular of degree (n − 1), disconnected with n − connected components and has a girth of 3. Lastly in part C, the graphs Γ(n−1)+2, Γ(n−1)+3, Γ(n−1)+4,..., Γ(n−1)+n are found to be undirected, regular of degree (n − 1)2, disconnected with n2− connected components and has a girth of 3. },
 year = {2024}
}

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TY  - JOUR
T1  - Suborbital Graphs Involving the Action of the Direct Product of Alternating and Cyclic Groups on the Cartesian Product of Two Sets
AU  - Morang’a Daniel Orina
AU  - Nyaga Lewis Namu
AU  - Gikunju David Muriuki
Y1  - 2024/10/31
PY  - 2024
N1  - https://doi.org/10.11648/j.engmath.20240801.12
DO  - 10.11648/j.engmath.20240801.12
T2  - Engineering Mathematics
JF  - Engineering Mathematics
JO  - Engineering Mathematics
SP  - 8
EP  - 17
PB  - Science Publishing Group
SN  - 2640-088X
UR  - https://doi.org/10.11648/j.engmath.20240801.12
AB  - The combinatorial properties and invariants which includes transitivity, primitivity, ranks and subdegrees of direct product of Alternating group and Cyclic group acting on Cartesian product of two set have been extensively studied. However, the construction of suborbital graphs for this group action remains largely unexplored. As a result, this research paper addresses this gap by constructing suborbital graphs involving direct product of the Alternating group and Cyclic group acting on the Cartesian product of two sets where their respective properties such as connectivity, self-pairedness, girth and vertex degree are analyzed in detail. First, the result shows that all suborbits are self − paired implies that the vertex sets are undirected to each other. Secondly, the constructed suborbital graphs are classified into three parts for any value of n ≥ 3. In part A, it is proven that the constructed suborbital graphs Γ1, Γ2, Γ3,..., Γ(n−1) are undirected, regular of degree (n−1), disconnected with n−connected components and has a girth of 3. In part B, the constructed suborbital graph Γ(n−1)+1 is found to be undirected, regular of degree (n − 1), disconnected with n − connected components and has a girth of 3. Lastly in part C, the graphs Γ(n−1)+2, Γ(n−1)+3, Γ(n−1)+4,..., Γ(n−1)+n are found to be undirected, regular of degree (n − 1)2, disconnected with n2− connected components and has a girth of 3. 
VL  - 8
IS  - 1
ER  -

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