AN ANALYSIS ON $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - REGULAR AND TOTAL REGULAR CUBIC FUZZY GRAPHS

J. Befija Minnie; L. Ancy Lara; D. Ajay

AN ANALYSIS ON $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - REGULAR AND TOTAL REGULAR CUBIC FUZZY GRAPHS

Abstract

This article introduces the concepts of $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - regular and total $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - regular cubic fuzzy graphs, extending fuzzy graph theory by applying regularity conditions to both vertex and edge membership functions. The study defines and analyzes the vertex $d_2$ - degree and Total $d_{2}$ - degree in cubic fuzzy graphs, including $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - regularity and Total $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - regularity to determine the conditions under which a cubic fuzzy graph is regular. Furthermore, $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - regularity is examined in cyclic cubic fuzzy graphs with specific membership values. Our results formalize new regularity conditions in cubic fuzzy graphs, providing a deeper understanding of their properties. The study further explores the regularity properties of cubic fuzzy graphs, providing new tools for potential applications in fields such as network analysis, decision-making and related fields.

Keywords

References

[1] Zadeh, L. A., (1965), Fuzzy sets, Information and control, 8(3), 338-353.
[2] Zadeh, L. A., (1968), Fuzzy algorithms, Information and Control, 12(2), 94-102.
[3] Zadeh, L. A., (1971), Similarity relations and fuzzy orderings, Information Sciences, 3(2), 177-200.
[4] Kauffman, A., (1973), A Introduction a la Theorie des Sous-Emsembles Flous, 1. French:Masson:Issy-les-Moulineaux.
[5] Zadeh, L. A., (1975), The concept of a linguistic variable and its application to approximate reasoning-I, Inf Sci, 8(3), 199-249.
[6] Rosenfeld, A., (1975), Fuzzy Graph, In: Zadeh, L. A., Fu, K. S., Shimura, M. EDs., Fuzzy Sets and Their Applications, Academic press, pp. 77-95.
[7] Bondy, J. A., Murty, U. S. R., (1979), Graph Theory with Applications, MacMillan, London.
[8] Bhattacharya, P., (1987), Some Remarks on Fuzzy Graphs, Pattern Recognition Lett., 6, 297-302.

[9] Balakrishnan, R., Ranganathan, K., (1991), A Text Book of Graph Theory, Springer.
[10] Moderson, J. N., Nair, P. S., (2000), Fuzzy Graphs and Fuzzy Hypergraphs, Physica-Verlag, Heidelberg.
[11] Nagoor Gani, A., Radha, K., (2008), On Regular Fuzzy Graphs, Journal of Physical Sciences, Vol. 12, 33 - 40.
[12] Jun, Y. B., Kim, C. S., Yang, K. O., (2012), Cubic sets, Ann. Fuzzy Math. Inform., 4(1), 1-32.
[13] Santhi Maheswari, N. R., Sekar, C., (2014), On (2, k) - regular fuzzy graph and totally (2, k) - regular fuzzy graph, International Journal of Mathematics and Soft computing, 4(2), 59-69.
[14] Borzooei, R., Rashmanlou, H., (2016), More Results on Vague Graphs, UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 78(1).
[15] Santhi Maheswari, N. R., Sekar, C., (2016), On (r, 2, k) - regular fuzzy graph, Journal of Combinatorics and Mathematicsal Combinatorial Computing, 97, pp. 11-21.
[16] Ravi Narayanan, S., Murugesan, S., (2016), On (r1, r2), 2, (c1, c2) - Pseudo Regular Intuitionistic Fuzzy Graphs, Advances in Fuzzy Sets and Systems, Volume 21, Number 4, Pages 265-280.
[17] Mathew, S., Mordeson, J. N., Malik, D. S., (2017) Fuzzy graph Theory, Springer.
[18] Rashid, S., Yaqoob. N., Akram, M., Gulistan, M., (2018), Cubic graphs with application, Int.J. Anal. Appl., 16(5), 733-750.
[19] Huda Mutab Al Mutab, (2019), Fuzzy Graphs, Journal of Advances In Mathematics, Vol 17
[20] Pal, M., Samanta, S., Ghorai, G., (2019), Modern Trends in Fuzzy Graph Theory, Singapore: Springer.
[21] Muhiuddin, G., Takallo, M. M., Jun, Y. B., Borzooei, R. A., (2020), Cubic graphs and their application to a traffic flow problem, Int.J.Comput. Intell. Syst., 13(1), 1265-1280.
[22] Shoaib, M., Kosari, S., Rashmanlou, H., Malik, M. A., Rao, Y., Talebi, Y., Mofidnakhaei, F., (2020), Notion of Complex Pythagorean Fuzzy Graph with Properties and Application, J. Multiple Valued Log. Soft Comput., 34, 553–586.
[23] Rashmanlou, H., Muhiuddin, G., Amanathulla, S. K., Mofidnakhaei, F., Pal, M., (2021), a study on cubic graphs with novel application, J. Intell. Fuzzy Syst., 40(1), 89-101.
[24] Amanathulla, S., Pal, M., (2022), An introduction to Picture Fuzzy Graph and its application to select best routes in an airlines network, In Advances in computer and electrical engineering book series, pp. 385–411.
[25] Jiang, H., Talebi, A. A., Shao, Z., Sadati, S. H., Rashmanlou, H., (2022), New concepts of vertex covering in cubic graphs with its applications, Mathematics, 10(3), 307.
[26] Muhiuddin, G., Hameed, S., Maryam, A., Ahmad, U., (2022), Cubic Pythagorean Fuzzy Graphs, Journal of Mathematics, 2022(1).
[27] Shi, X., Akhoundi, M., Talebi, A. A., Sadati, S. H., (2022), Some Properties of Cubic Fuzzy Graphs with an Application, Symmetry, 14(12), 2623.
[28] Khatun, J., Amanathulla, S., Pal, M., (2023), Picture fuzzy cubic graphs and their applications, Journal of Intelligent and Fuzzy Systems, 46(1), 2981–2998.
[29] Kosari, S., Jiang, H., Khan, A., Akhoundi, M., (2023), Properties of Connectivity in Vague Fuzzy Graphs With Application in Building University, Journal of Multiple-valued Logic and Soft Computing, 41, 463-482.
[30] Li, L., Kosari, S., Sadati, S. H., Talebi, A. A., (2023), Concepts of vertex regularity in cubic fuzzy graph structures with an application, Frontiers in Physics, 10.
[31] Lu, J., Zhu, L., Gao, W., (2023), Remarks on bipolar cubic fuzzy graphs and its chemical applications, International Journal of Mathematics and Computer in Engineering, 1(1), 1–10.
[32] Shi, X., Kosari, S., Sadati, S. H., Talebi, A. A., Khan, A., (2023), Special concepts of edge regularity in the cubic fuzzy graph structure environment with an application, Frontiers in Physics, 11.
[33] Banerjee, A., Amanathulla, S., (2024), Optimization of disaster management using split domination in picture fuzzy graphs, Journal of Applied Mathematics and Computing, 70(1), 435–459.
[34] Guan, H., Sadati, S. H., Talebi, A. A., Shafi, J., Khan, A., (2024), A connectivity index based on adjacent vertices in cubic fuzzy graph with an application, Journal of Intelligent and Fuzzy Systems, 46(4), 11025–11040.
[35] Khatun, J., Amanathulla, S., Pal, M., (2024), A comprehensive study on m-polar picture fuzzy graphs and its application, Journal of Uncertain Systems, Vol. 18, No. 01, 2450016.
[36] Kosari, S., Shi, X., Kacprzyk, J., Chen, Z., Rashmanlou, H., (2024), A Novel Description of Perfectly Regular Fuzzy Graphs with Application in Psychological Sciences, Journal of Multiple-valued Logic and Soft Computing, 42, 405-424.
[37] Kosari, S., Qiang, X., Kacprzyk, J., Ain, Q., Rashmanlou, H., (2024), A Study on Topological Indices in Fuzzy Graphs with Application in Decision Making Probelms, Journal of Multiple-valued Logic and Soft Computing, 42(5-6), 567-589.
[38] Maryam, A., Rahim, M. T., Hussain, F., (2024), Properties of Cubic Fuzzy Competition Graphs with Applications, Ars Combinatoria, 160(1), 141–159.
[39] Shi, X., Kosari, S., Hameed, S., Shah, A. G., Ullah, S., (2024), Application of connectivity index of cubic fuzzy graphs for identification of danger zones of tsunami threat, PLoS ONE 19(1), e0297197.
[40] Bera, B., Banerjee, A., Khatun, J., Amanathulla, S., Pal, M., (2025), Some properties of regular picture fuzzy graph with application in social network analysis, Journal of Applied Mathematics and Computing, Volume 71, pages 2959–2982.
[41] Hong, S., Yue, T., You, Y., Lv, Z., Tang, X., Hu, J., Yin, H., (2025), A resilience recovery method for complex traffic network security based on trend forecasting, International Journal of Intelligent Systems, 2025(1).
[42] Kosari, S., Shao, Z., Raman, S., Ganesan, B., (2025), A study on strong and geodetic domination integrity sets in graphs, Communications in Combinatorics and Optimization, (Accepted).
[43] Rao, Y., Kosari, S., Hameed, S., Yousaf, Z., (2025), Multi-attribute decision-making using q-rung orthopair fuzzy Zagreb index, Artificial Intelligence Review, 58(153).
[44] Xu, J., Liu, W., Zhang, K., Zhu, E., (2025), Dna coding theory and algorithms, Artificial Intelligence Review, 58(6).
[45] Kolandasamy, R., Ramesh, D., Iampan, A., Rao, G. S., Jayanti, S., Abdulkadhar, S. (2025), Regular Fuzzy Graphs with Chromatic Numbers, International Journal of Analysis and Applications, 23, 173.
[46] Jaisankar, D., Ramalingam, S., Zegeye, G. B. (2025), A fuzzy graph theoretic approach to face shape recognition using cubic outerplanar structures, Scientific Reports, 15(1), 44072.

Details

Primary Language

English

Subjects

Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics), Mathematical Logic, Set Theory, Lattices and Universal Algebra

Journal Section

Research Article

Authors

J. Befija Minnie ^* This is me
0000-0001-7628-1388
India

L. Ancy Lara This is me
0009-0007-8987-284X
India

D. Ajay
0000-0001-8890-4626
India

Publication Date

August 7, 2026

Submission Date

May 22, 2025

Acceptance Date

October 14, 2025

Published in Issue

Year 2026 Volume: 16 Number: 8

IZ

https://izlik.org/JA28NS24CT

Cite

RIS / Bibtex

APA

Minnie, J. B., Lara, L. A., & Ajay, D. (2026). AN ANALYSIS ON $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - REGULAR AND TOTAL REGULAR CUBIC FUZZY GRAPHS. TWMS Journal of Applied and Engineering Mathematics, 16(8), 964-977. https://izlik.org/JA28NS24CT

AMA

1.Minnie JB, Lara LA, Ajay D. AN ANALYSIS ON $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - REGULAR AND TOTAL REGULAR CUBIC FUZZY GRAPHS. JAEM. 2026;16(8):964-977. https://izlik.org/JA28NS24CT

Chicago

Minnie, J. Befija, L. Ancy Lara, and D. Ajay. 2026. “AN ANALYSIS ON $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - REGULAR AND TOTAL REGULAR CUBIC FUZZY GRAPHS”. TWMS Journal of Applied and Engineering Mathematics 16 (8): 964-77. https://izlik.org/JA28NS24CT.

EndNote

Minnie JB, Lara LA, Ajay D (August 1, 2026) AN ANALYSIS ON $([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}) $ - REGULAR AND TOTAL REGULAR CUBIC FUZZY GRAPHS. TWMS Journal of Applied and Engineering Mathematics 16 8 964–977.

IEEE

[1]J. B. Minnie, L. A. Lara, and D. Ajay, “AN ANALYSIS ON $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - REGULAR AND TOTAL REGULAR CUBIC FUZZY GRAPHS”, JAEM, vol. 16, no. 8, pp. 964–977, Aug. 2026, [Online]. Available: https://izlik.org/JA28NS24CT

ISNAD

Minnie, J. Befija - Lara, L. Ancy - Ajay, D. “AN ANALYSIS ON $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - REGULAR AND TOTAL REGULAR CUBIC FUZZY GRAPHS”. TWMS Journal of Applied and Engineering Mathematics 16/8 (August 1, 2026): 964-977. https://izlik.org/JA28NS24CT.

JAMA

1.Minnie JB, Lara LA, Ajay D. AN ANALYSIS ON $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - REGULAR AND TOTAL REGULAR CUBIC FUZZY GRAPHS. JAEM. 2026;16:964–977.

MLA

Minnie, J. Befija, et al. “AN ANALYSIS ON $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - REGULAR AND TOTAL REGULAR CUBIC FUZZY GRAPHS”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 8, Aug. 2026, pp. 964-77, https://izlik.org/JA28NS24CT.

Vancouver

1.J. Befija Minnie, L. Ancy Lara, D. Ajay. AN ANALYSIS ON $(([r_{\alpha},r_{\beta}],r_{\gamma}), 2, ([k_{\alpha},k_{\beta}],k_{\gamma}))$ - REGULAR AND TOTAL REGULAR CUBIC FUZZY GRAPHS. JAEM [Internet]. 2026 Aug. 1;16(8):964-77. Available from: https://izlik.org/JA28NS24CT