An application on edge irregular reflexive labeling for $ m^t $-graph of cycle graph

Muhammad Amir Asif; Rashad Ismail; Ayesha Razaq; Esmail Hassan Abdullatif Al-Sabri; Muhammad Haris Mateen; Shahbaz Ali; Muhammad Amir Asif; Rashad Ismail; Ayesha Razaq; Esmail Hassan Abdullatif Al-Sabri; Muhammad Haris Mateen; Shahbaz Ali

doi:10.3934/math.2025060

AIMS Mathematics

2025, Volume 10, Issue 1: 1300-1321. doi: 10.3934/math.2025060

Previous Article Next Article

Research article

An application on edge irregular reflexive labeling for $ m^t $-graph of cycle graph

1.
School of Mathematics, Minhaj University, Lahore, Pakistan
2.
Department of Mathematics, Faculty of Science and Arts, Mahayl Assir, King Khalid University, Abha, Saudi Arabia
3.
Department of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Pakistan

Received: 28 October 2024 Revised: 08 January 2025 Accepted: 10 January 2025 Published: 21 January 2025
MSC : 05C12, 05C90

Graph labeling is an increasingly popular problem in graph theory. A mapping converts a collection of graph components into a set of integers known as labels. Graph labeling techniques typically label edges with positive integers, vertices with even numbers, and edge weights with consecutive numbers, known as edge irregular reflexive total labeling. This is achieved by utilizing the reflexive edge irregularity strength of the graphical structure. The edge calculates the exact values of the reflexive edge irregularity strength irregular reflexive labeling for the $ m^t $-graph of cycle graph $ mC_n $ on $ t = 1 $ with $ n \geq 3 $ and $ m \geq 4 $. The maximum number of assignments assigned to each individual in a communication network, as well as providing a secure communication channel to ensure the unique identification of each employee, are potential applications for this problem.
- reflexive edge irregularity strength,
- $ m^t $-graphs,
- edge irregular reflexive total labeling
Citation: Muhammad Amir Asif, Rashad Ismail, Ayesha Razaq, Esmail Hassan Abdullatif Al-Sabri, Muhammad Haris Mateen, Shahbaz Ali. An application on edge irregular reflexive labeling for $ m^t $-graph of cycle graph[J]. AIMS Mathematics, 2025, 10(1): 1300-1321. doi: 10.3934/math.2025060

Related Papers:

Abstract

Graph labeling is an increasingly popular problem in graph theory. A mapping converts a collection of graph components into a set of integers known as labels. Graph labeling techniques typically label edges with positive integers, vertices with even numbers, and edge weights with consecutive numbers, known as edge irregular reflexive total labeling. This is achieved by utilizing the reflexive edge irregularity strength of the graphical structure. The edge calculates the exact values of the reflexive edge irregularity strength irregular reflexive labeling for the $ m^t $-graph of cycle graph $ mC_n $ on $ t = 1 $ with $ n \geq 3 $ and $ m \geq 4 $. The maximum number of assignments assigned to each individual in a communication network, as well as providing a secure communication channel to ensure the unique identification of each employee, are potential applications for this problem.

References

[1]	A. Naumowicz, A note on the seven bridges of Königsberg problem, Formal. Math., 22 (2014), 177–178. http://doi.org/10.2478/forma-2014-0018 doi: 10.2478/forma-2014-0018
[2]	W. Gao, H. Wu, M. K. Siddiqui, A. Q. Baig, Study of biological networks using graph theory, Saudi J. Biol. Sci., 25 (2018), 1212–1219. https://doi.org/10.1016/j.sjbs.2017.11.022 doi: 10.1016/j.sjbs.2017.11.022
[3]	N. Deo, Graph theory with applications to engineering and computer science, New York: Courier Dover Publications, 2017.
[4]	E. Estrada, Graph and network theory, Glasgow: University of Strathclyde, 2013.
[5]	K. M. M. Haque, Irregular total labelings of generalized Petersen graphs, Theory Comput. Syst., 50 (2012), 537–544. https://doi.org/10.1007/s00224-011-9350-7 doi: 10.1007/s00224-011-9350-7
[6]	G. Chartrand, M. S. Jacobson, J. Lehel, O. R. Oellermann, S. Ruiz, F. Saba, Irregular networks, Congr. Numer., 64 (1988), 197–210.
[7]	A. Ahmad, O. B. S. Al-Mushayt, M. Bača, On edge irregularity strength of graphs, Appl. Math. Comput., 243 (2014), 607–610. https://doi.org/10.1016/j.amc.2014.06.028 doi: 10.1016/j.amc.2014.06.028
[8]	M. Imran, A. Q. Baig, M. Town, Vertex irregular total labeling of cubic graphs, Util. Math., 91 (2013), 287–299.
[9]	D. Indriati, I. E. Wijayanti, K. A. Sugeng, On total irregularity strength of double-star and related graphs, Procedia Comput. Sci., 74 (2015), 118–123. https://doi.org/10.1016/j.procs.2015.12.086 doi: 10.1016/j.procs.2015.12.086
[10]	D. Indriati, On vertex irregular total k-labeling and total vertex irregularity strength of lollipop graphs, J. Phys.: Conf. Ser., 1306 (2019), 012025. https://doi.org/10.1088/1742-6596/1306/1/012025 doi: 10.1088/1742-6596/1306/1/012025
[11]	S. Jendrol', J. Miškuf, R. Soták, Total edge irregularity strength of complete graphs and complete bipartite graphs, Discr. Math., 310 (2010), 400–407. https://doi.org/10.1016/j.disc.2009.03.006 doi: 10.1016/j.disc.2009.03.006
[12]	M. Bača, M. K. Siddiqui, Total edge irregularity strength of generalized prism, Appl. Math. Comput., 235 (2014), 168–173. https://doi.org/10.1016/j.amc.2014.03.001 doi: 10.1016/j.amc.2014.03.001
[13]	M. Bača, M. Miller, J. Ryan, On irregular total labellings, Discr. Math., 307 (2007), 1378–1388. https://doi.org/10.1016/j.disc.2005.11.075 doi: 10.1016/j.disc.2005.11.075
[14]	J. Ivančo, S. Jendrol', Total edge irregularity strength of trees, Discus. Math. Graph Theory, 26 (2006), 449–456.
[15]	M. Ibrahim, S. Majeed, M. K. Siddiqui, Edge irregular reflexive labeling for star, double star and caterpillar graphs, TWMS J. Appl. Eng. Math., 10 (2020), 718–726.
[16]	G. Xin, X. Xu, C. Zhang, Y. Zhong, On magic distinct labellings of simple graphs, J. Symbol. Comput., 119 (2023), 22–37. https://doi.org/10.1016/j.jsc.2023.02.005 doi: 10.1016/j.jsc.2023.02.005
[17]	Y. C. Liang, X. Zhu, Anti-magic labelling of Cartesian product of graphs, Theor. Comput. Sci., 477 (2013), 1–5. https://doi.org/10.1016/j.tcs.2012.12.023 doi: 10.1016/j.tcs.2012.12.023
[18]	N. Hartsfield, G. Ringel, Pearls in graph theory: A comprehensive introduction, North Chelmsford: Courier Corporation, 2013.
[19]	A. Lladó, J. Moragas, Cycle-magic graphs, Discr. Math., 307 (2007), 2925–2933. https://doi.org/10.1016/j.disc.2007.03.007 doi: 10.1016/j.disc.2007.03.007
[20]	M. K. Mahmood, S. Ali, A novel labeling algorithm on several classes of graphs, Punjab Univ. J. Math., 49 (2017), 23–35.
[21]	M. K. Mahmood, S. Ali, On super totient numbers, with applications and algorithms to graph labeling, Ars Comb., 143 (2019), 29–37.
[22]	S. Ali, K. Mahmood, New numbers on Euler's totient function with applications, J. Math. Extens., 14 (2019), 61–83.
[23]	A. Shahbaz, M. K. Mahmood, K. P. Shum, Novel classes of integers and their applications in graph labeling, Hacet. J. Math. Stat., 50 (2021), 1094–1110. https://doi.org/10.15672/hujms.825723 doi: 10.15672/hujms.825723
[24]	A. S. Alali, S. Ali, N. Hassan, A. M. Mahnashi, Y. Shang, A. Assiry, Algebraic structure graphs over the commutative ring Z m: exploring topological indices and entropies using M-polynomials, Mathematics, 11 (2023), 3833. https://doi.org/10.3390/math11183833 doi: 10.3390/math11183833
[25]	S. Ali, M. K. Mahmmod, R. M. Falcón Ganfornina, paradigmatic approach to investigate restricted hyper totient graphs, AIMS Math., 6 (2021), 3761–3771. https://doi.org/10.3934/math.2021223 doi: 10.3934/math.2021223
[26]	S. Ali, M. K. Mahmood, A paradigmatic approach to investigate restricted totient graphs and their indices, Comput. Sci., 16 (2021), 793–801.
[27]	M. H. Mateen, M. K. Mahmmod, D. A. Kattan, S. Ali, A novel approach to find partitions of Zm with equal sum subsets via complete graphs, AIMS Math., 6 (2021), 9998–10024. http://dx.doi.org/10.3934/math.2021581 doi: 10.3934/math.2021581
[28]	S. V. Gervacio, Cycle graphs, In: Graph Theory Singapore 1983: Proceedings of the First Southeast Asian Graph Theory Colloquium, 1984,279–293.
[29]	A. Ayache, A. Alameri, Topological indices of the mk-graph, J. Assoc. Arab Uni. Basic Appl. Sci., 24 (2017), 283–291. https://doi.org/10.1016/j.jaubas.2017.03.001 doi: 10.1016/j.jaubas.2017.03.001
[30]	Y. Ke, M. J. A. Khan, M. Ibrahim, M. K. Siddiqui, On edge irregular reflexive labeling for Cartesian product of two graphs, Eur. Phys. J. Plus, 136 (2021), 6. https://doi.org/10.1140/epjp/s13360-020-00960-1 doi: 10.1140/epjp/s13360-020-00960-1
[31]	S. Jendrol', Total labellings, irregularity strengths and colourings, AKCE Int. J. Graphs Comb., 6 (2009), 219–227.
[32]	A. Ahmad, M. Arshad, G. Ižaríková, Irregular labelings of helm and sun graphs, AKCE Int. J. Graphs Comb., 12 (2015), 161–168. https://doi.org/10.1016/j.akcej.2015.11.010 doi: 10.1016/j.akcej.2015.11.010
[33]	D. Tanna, J. Ryan, A. Semaničová-Feňovčíková, Edge irregular reflexive labeling of prisms and wheels, Australas. J. Comb., 69 (2017), 394–401.
[34]	M. J. A. Khan, M. Ibrahim, A. Ahmad, On edge irregular reflexive labeling of categorical product of two paths, Comput. Syst. Sci. Eng., 36 (2021), 485–492. http://doi.org/10.32604/csse.2021.014810 doi: 10.32604/csse.2021.014810
[35]	C. Wang, M. J. A. Khan, M. Ibrahim, E. Bonyah, M. K. Siddiqui, S. Khalid, On edge irregular reflexive labeling for generalized prism, J. Math., 2022 (2022), 2886555. https://doi.org/10.1155/2022/2886555 doi: 10.1155/2022/2886555
[36]	M. Bača, M. Irfan, J. Ryan, A. Semaničová-Feňovčíková, D Tanna, Note on edge irregular reflexive labelings of graphs, AKCE Int. J. Graphs Comb., 16 (2019), 145–157. https://doi.org/10.1016/j.akcej.2018.01.013 doi: 10.1016/j.akcej.2018.01.013
[37]	M. Basher, On the reflexive edge strength of the circulant graphs, AIMS Math., 6 (2021), 9342–9365. http://doi.org/10.3934/math.2021543 doi: 10.3934/math.2021543
[38]	I. H. Agustin, M. I. Utoyo, M. Venkatachalam, The reflexive edge strength on some almost regular graphs, Heliyon, 7 (2021), e06991.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)