First Zagreb Index and its Characteristics on Neutrosophic Graph

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Published: Jul 30, 2025
DOI: https://doi.org/10.33093/ijoras.2025.7.2.3
Keywords:
Topological Index, Zagreb Index of First Type, Neutrosophic Set, Complement of Pythagorean Co-Neutrosophic Graph, Neutrosophic Zagreb Index of First Type, Neutrosophic Entire Zagreb Index of First Type

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Govindan Vetrivel
Department of Mathematics, Alagappa University (India)
https://orcid.org/0000-0001-9375-4933
Murugappan Mullai
Department of Mathematics, Alagappa University (India)
https://orcid.org/0000-0001-5762-1308
Greinggrai Rajchakit
Faculty of Science, Maejo University (Thailand)
https://orcid.org/0000-0001-6053-6219
Meyyappan Sangavi
Department of Mathematics, Alagappa University (India)
https://orcid.org/0009-0005-7286-4945
R. Surya
Department of Mathematics, Alagappa University (India)
https://orcid.org/0000-0002-5671-6417

Abstract

Topological indices mark an irreplaceable place for applications in crisp and fuzzy graphs. These indices are extended to the neutrosophic graphical idea to rectify the imprecise values or information acquired before, since the uncertain cases are organized and allocated as a separate membership called "indeterminacy". We apply and explore the First zagreb index and its properties on the neutrosophic graphical system in the line of Wiener and Forgotten indices. This fills the gap between fuzzy and its graphical extensions on indices discussion, thereby extends the applicable areas. Also, an improvised and unique application is portrayed to observe the importance of First zagreb index in the neutrosophic theme of graphs. This contributes to the real life in a greater way than the fuzzy idea.


 


Manuscript received:8 Apr 2025 | Revised: 31 May 2025 | Accepted: 19 Jun 2025 | Published: 30 Jul 2025

Article Details

How to Cite
Govindan, V., Murugappan, M., Rajchakit, G., Meyyappan, S., & R, S. . (2025). First Zagreb Index and its Characteristics on Neutrosophic Graph. International Journal on Robotics, Automation and Sciences, 7(2), 22–28. https://doi.org/10.33093/ijoras.2025.7.2.3
Issue
Vol. 7 No. 2 (2025): International Journal on Robotics, Automation and Sciences
Section
Articles
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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Author Biographies

Govindan Vetrivel, Department of Mathematics, Alagappa University (India)

He is a research scholar in Department of Mathematics

Greinggrai Rajchakit, Faculty of Science, Maejo University (Thailand)

He is working as an Associate Professor in Mathematics, Maejo University, Thailand

Meyyappan Sangavi, Department of Mathematics, Alagappa University (India)

She is a research scholar in Department of Mathematics

R. Surya, Department of Mathematics, Alagappa University (India)

She has completed her Ph.D in Alagappa University

References

L.A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338-353, 1965.

DOI: https://doi.org/10.1016/S0019-9958(65)90241-X

A. Rosenfeld, “Fuzzy graphs”, Fuzzy Sets and Their

Applications to Cognitive and Decision Processes, Academic press, 1975.

DOI: https://doi.org/10.1016/B978-0-12-775260-0.50008-6

A. Nagoor Gani and D. Rajalaxmi (a) Subahashini, “Fuzzy Labeling Tree,” International Journal of Pure and Applied Mathematics, vol. 90, no. 2, pp. 131-141, 2014.

DOI: http://dx.doi.org/10.12732/ijpam.v90i2.3

S. Kalathian, S. Ramalingam, S. Raman and N. Srinivasan, “Some topological indices in fuzzy graphs,” Journal of Intelligent and Fuzzy Systems, vol. 39, no. 5, pp. 6033-6046,

DOI: https://doi.org/10.3233/JIFS-189077

S. Poulik and G. Ghorai, “Certain indices of graphs under bipolar fuzzy environment with applications,” Soft Computing, vol. 24, pp. 5119-5131, 2020.

DOI: https://doi.org/10.1007/s00500-019-04265-z

S. Poulik and G. Ghorai, “Determination of journeys order based on graph’s Wiener absolute index with bipolar fuzzy information,” Information Sciences, vol. 545, pp. 608–619, 2021.

DOI: https://doi.org/10.1016/j.ins.2020.09.050

S. Poulik, S. Das, and G. Ghorai, “Randic index of bipolar fuzzy graphs and its application in network systems,” Journal of Applied Mathematics and Computing, vol. 68, pp. 2317–2341, 2022.

DOI: https://doi.org/10.1007/s12190-021-01619-5

S.R. Islam and M. Pal, “F-index for fuzzy graph with application,” TWMS Journal of Applied and Engineering Mathematics, vol. 13, no. 2, pp. 517-530, 2023.

URL: https://jaem.isikun.edu.tr/web/images/articles/vol.13.no.2/11.pdf

S.R. Islam and M. Pal, “Further development of F-index for fuzzy graph and its application in Indian railway crime,” Journal of Applied Mathematics and Computing, vol. 69, pp. 321–353, 2023.

DOI: https://doi.org/10.1007/s12190-022-01748-5

S.R. Islam and M. Pal, “An investigation of edge F-index on fuzzy graphs and application in molecular chemistry,” Complex & Intelligent Systems, 9, pp. 2043–2063, 2023.

DOI: https://doi.org/10.1007/s40747-022-00896-2

S.R. Islam and M. Pal, “First Zagreb index on a fuzzy graph and its application,” Journal of Intelligent and Fuzzy Systems, vol. 40, no. 6, pp. 10575-10587, 2021.

DOI: https://doi.org/10.3233/JIFS-201293

S.R. Islam and M. Pal, “Second Zagreb index for fuzzy graphs and its application in mathematical chemistry,” Iranian Journal of Fuzzy Systems, vol. 20, no. 1, pp. 119-136, 2023.

DOI: https://doi.org/10.22111/ijfs.2023.7350

S.R. Islam and M. Pal, “Hyper-Wiener Index for Fuzzy Graph and Its Application in Share Market,” Journal of Intelligent and Fuzzy Systems, vol. 41, no. 1, pp. 2073-2083, 2021.

DOI: https://doi.org/10.3233/JIFS-210736

S.R. Islam, B.B. Mohsin and M. Pal, “Hyper-Zagreb index in fuzzy environment and its application,” Heliyon, vol. 10, no. 16, pp. 2024.

DOI: https://doi.org/10.1016/j.heliyon.2024.e36110

U. Ahmad, N.K. Khan and A.B. Saeid, “Fuzzy topological indices with application to cybercrime problem,” Granular Computing, vol. 8, pp. 967–980, 2023.

DOI: https://doi.org/10.1007/s41066-023-00365-2

U. Jana and G. Ghorai, “First Entire Zagreb Index of Fuzzy Graph and Its Application,” Axioms, vol. 12, no. 5, 2023.

DOI: https://doi.org/10.3390/axioms12050415

U. Jana and G. Ghorai, “Gutman index of fuzzy graphs with application,” Social Network Analysis and Mining, vol. 14, no. 209, 2024.

DOI: https://doi.org/10.1007/s13278-024-01358-y

S.I. Abdullah, S. Samanta, K. De et al. “Properties of the forgotten index in bipolar fuzzy graphs and applications,” Scientific Reports, vol. 14, no. 28264, 2024.

DOI: https://doi.org/10.1038/s41598-024-79295-1

M.U. Mirza, R. Anjum, M.Z. Youssef and T. Alsuraiheed. “A comprehensive study on fuzzy and crisp graph indices: generalized formulae, proximity and accuracy analysis,” AIMS Mathematics, vol. 8, no. 12, pp. 30922-30939, 2023.

DOI: https://doi.org/10.3934/math.20231582

N. Sarala and R. Abirami, “On Topological Indices of Fuzzy Random Graph.” Indian Journal of Science and Technology, vol. 17, no. 30, pp. 3080-3092, 2024.

DOI: https://doi.org/10.17485/IJST/v17i30.1695

K.T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87-96, 1986.

DOI: https://doi.org/10.1016/S0165-0114(86)80034-3

R. Parvathi and M.G. Karunambigai, “Intuitionistic Fuzzy Graphs,” Computational Intelligence, Theory and Applications, vol. 38, Springer, pp. 139-150, 2006.

DOI: https://doi.org/10.1007/3 540-34783-6_15

M. Akram and R. Akmal, “Operations on intuitionistic fuzzy graph structures,” Fuzzy Information and Engineering, vol. 8, no. 4, pp. 389-410, 2016.

DOI: https://doi.org/10.1016/j.fiae.2017.01.001

Javeria Dinar, Zahid Hussain, Shahid Zaman, Shams Ur Rehman, “Wiener index for an intuitionistic fuzzy graph and its application in water pipeline network,” Ain Shams

Engineering Journal, vol. 14, no. 1, 2023.

DOI: https://doi.org/10.1016/j.asej.2022.101826

M. Imran, M. Azeem, M.K. Jamil et al., “Some operations on intuitionistic fuzzy graphs via novel versions of the sombor index for internet routing,” Granular Computing, vol. 9, no. 53,

DOI: https://doi.org/10.1007/s41066-024-00467-5

M. Kamran Jamil, S. Anwer, M. Azeem and I. Gutman, "Intuitionistic fuzzy Sombor indices: A novel approach for improving the performance of vaccination centers," Communications in Combinatorics and Optimization, vol. 10, no. 3, pp. 563-593, 2025.

DOI: https://doi.org/10.22049/cco.2023.28767.1709

N. Shaik and S.B. Shaik, “Wiener index application in intuitionistic fuzzy rough graphs for transport network flow,” Scientific Reports, vol. 15, no. 9591, pp. 1-21, 2025.

DOI: https://doi.org/10.1038/s41598-025-94488-y

F. Smarandache, “Neutrosophic set- a generalization of the intuitionistic fuzzy set,” 2006 IEEE International Conference on Granular Computing, pp. 38-42, 2006.

DOI: https://doi.org/10.1109/GRC.2006.1635754

F. Smarandache, “A geometric interpretation of the neutrosophic set- A generalization of the intuitionistic fuzzy set,” 2011 IEEE International Conference on Granular Computing, pp. 602-606, 2011.

DOI: https://doi.org/10.1109/GRC.2011.6122665

W.B. Vasantha Kandasamy, K. Ilanthenral, F. Smarandache, Neutrosophic graphs: A New Dimension to Graph Theory, Belgium: EuropaNova ASBL, Brussels, 2015.

URL: https://digitalrepository.unm.edu/math_fsp/98/

S. Broumi, F. Smarandache, M. Talea and A. Bakali, “Single valued neutrosophic graphs: Degree, order and size,” 2016 IEEE International Conference on Fuzzy Systems (FUZZ-

IEEE), pp. 2444-2451, 2016.

DOI: https://doi.org/10.1109/FUZZ-IEEE.2016.7738000

M. Ghods and Z. Rostami, “Wiener index and applications in the Neutrosophic graphs,” Neutrosophic Sets Systems., vol. 46, pp. 229-245, 2021.

DOI: https://doi.org/10.5281/zenodo.5553515

Amir S. Majeed, Nabeel E. Arif, “Topological indices of certain neutrosophic graphs,” AIP Conference Proceedings, vol. 2845, no.1, p. 060024, 2023.

DOI: https://doi.org/10.1063/5.0157832

G. Vetrivel, M. Mullai, and R. Buvaneshwari, “Forgotten Topological Index and Its Properties on Neutrosophic Graphs,” Neutrosophic Systems With Applications, vol. 21,

pp. 36-45, 2024.

DOI: https://doi.org/10.61356/j.nswa.2024.21356

M. Kaviyarasu, M. Aslam, F. Afzal et al., “The connectivity indices concept of neutrosophic graph and their application of computer network, highway system and transport network

flow,” Scientific Reports, vol. 14, no. 4891, 2024.

DOI: https://doi.org/10.1038/s41598-024-54104-x