Sains Malaysiana 55(10)(2026): 1610-1617

http://doi.org/10.17576/jsm-2026-5510-01

 

Zagreb Index of Fuzzy Prime Coprime Graph of Integers Modulo Groups

(Indeks Zegreb daripada Graf Koprime Perdana Kabur bagi Kumpulan Integer Modulo)

 

ABDURAHIM, MAMIKA UJIANITA ROMDHINI*, I GEDE ADHITYA WISNU WARDHANA, FARIZ MAULANA & GUSTI YOGANANDA KARANG

Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Mataram, Mataram 83125, Indonesia

 

Diserahkan: 27 Julai 2025/Diterima: 10 September 2026

 

*Pengarang untuk surat-menyurat; email: mamika@unram.ac.id

Abstract

This study investigates the structural properties and topological indices of fuzzy prime coprime graphs constructed from integer modulo groups. A fuzzy prime coprime graph generalizes the conventional prime coprime graph by assigning membership values in the interval [0,1] to both vertices and edges, reflecting varying degrees of association. Unlike previous studies that used arbitrary assignments, this research introduces a systematic and mathematically rigorous method of weighting based on the greatest common divisor (gcd) and least common multiple (lcm) of the orders of group elements. The main contributions include the formulation of membership functions, the classification of vertex and edge types, and the derivation of closed-form expressions for the first Zagreb, second Zagreb, and entire Zagreb indices of the fuzzy graph. These results provide new insights into the interaction between fuzzy set theory and algebraic graph structures, promoting advances in mathematical modeling, fuzzy logic systems, and computational methods for complex networks. The approach also supports future research in discrete mathematics, data-driven systems, and symbolic computation.

Keywords: Fuzzy prime coprime graph; group ; Zagreb index

 

Abstrak

Penyelidikan ini mengkaji sifat struktur dan indeks topologi bagi graf koprime perdana kabur yang dibina daripada kumpulan modulo integer. Graf koprima perdana kabur menyamaratakan graf koprima perdana konvensional dengan memberikan nilai keahlian dalam selang [0,1] kepada kedua-dua bucu dan tepi, mencerminkan darjah perkaitan yang berbeza-beza. Tidak seperti kajian terdahulu yang menggunakan tugasan sewenang-wenangnya, penyelidikan ini memperkenalkan kaedah pemberat yang sistematik dan teliti secara matematik berdasarkan pembahagi sepunya terbesar (gcd) dan gandaan sepunya terkecil (lcm) bagi susunan unsur kumpulan. Sumbangan utama termasuk perumusan fungsi keahlian, pengkelasan jenis bucu dan tepi dan terbitan ungkapan bentuk tertutup untuk Zagreb pertama, Zagreb kedua dan keseluruhan indeks Zagreb bagi graf kabur. Keputusan ini memberikan pandangan baharu tentang interaksi antara teori set kabur dan struktur graf algebra, menggalakkan kemajuan dalam pemodelan matematik, sistem logik kabur dan kaedah pengiraan untuk rangkaian kompleks. Pendekatan ini juga menyokong penyelidikan masa depan dalam matematik diskret, sistem dipacu data dan pengiraan simbolik.

Kata kunci: Graf koprime perdana kabur; indeks Zagreb; kumpulan

RUJUKAN

Abdurahim, Romdhini, M.U. & Husain, S.K.S. 2025. Zagreb-based indices of prime coprime graph for integers modulo power of primes. Malaysian Journal of Mathematical Sciences 19(3): 961-974.

Abdurahim, Romdhini, M.U., Al-Sharqi, F. & Robbaniyyah, N.A. 2025. Relative prime coprime graph of integers modulo group and its reverse topological indices. Pan American Journal of Mathematics 4(10): 1-9.

Abdurahim, Pratiwi, L.F., Karang, G.Y., Wardhana, I.G.A.W., Irwansyah, Awanis, Z.Y. & Romdhini, M.U. 2026. Indeks Wiener pada graf koprima prima dari grup bilangan bulat modulo. Jurnal Matematika UNAND 15(2): 249-258.

Adhikari, A. & Banerjee, S. 2022. Prime coprime graph of a finite group. Novi Sad Journal of Mathematics 52(2): 41-59.

Afdhaluzzikri, M., Wardhana, I.G.A.W., Maulana, F. & Biswas, H.R. 2025. The non-coprime graphs of upper unitriangular matrix groups over the ring of integer modulo with prime order and their topological indices. BAREKENG: Jurnal Ilmu Matematika dan Terapan 19(1): 547-556.

Al-Mutab, H.M. 2019. Fuzzy graphs. Journal of Advances in Mathematics 17: 232-247.

Fayazi, F., Mahmudi, F. & Gholami, A. 2020. Some properties of fuzzy star graph and fuzzy line graph. Jordan Journal of Mathematics and Statistics 3(1): 139-151.

Hao, S., Zhong, G. & Ma, X. 2022. Notes on the co-prime order graph of a group. Proceedings of the Bulgarian Academy of Sciences 75(3): 340-348.

Islam, S.R. & Pal, M. 2023. Second Zagreb index for fuzzy graphs and its application in mathematical chemistry. Iranian Journal of Fuzzy Systems 20(1): 119-136.

Islam, S.R. & Pal, M. 2021. First Zagreb index on a fuzzy graph and its application. Journal of Intelligent & Fuzzy Systems 40(6): 10575-10587.

Jana, U. & Ghorai, G. 2023. First entire Zagreb index of fuzzy graph and its application. Axioms 12(5): 415.

Kalathian, S., Ramalingam, S., Raman, S. & Srinivasan, N. 2020. Some topological indices in fuzzy graphs. Journal of Intelligent and Fuzzy Systems 39(5): 6033-6046.

Mufti, Z.S., Tabraiz, A., Xin, Q., Almutairi, B. & Anjum, R. 2023a. Fuzzy topological analysis of pizza graph. AIMS Mathematics 8(6): 12841-12856.

Mufti, Z.S., Anjum, R., Tawfiq, F.M.O., Tabraiz, A., Xin, Q. & Akter, D. 2023b. Fuzzy topological characterization of qC_n graph via fuzzy topological indices. Journal of Mathematics 2023: 5141179.

Nurhabibah, N., Wardhana, I.G.A.W. & Switrayni, N.W. 2023. Numerical invariants of coprime graph of a generalized quaternion group. Journal of the Indonesian Mathematical Society 29(1): 36-44.

Romdhini, M.U., Abdurahim, Maharani, A.E.S.H. & Kamali, S.R. 2025a. Transmission-based energies of prime coprime graph for integers modulo group. Science and Technology Indonesia 10(3): 759-765.

Romdhini, M.U., Abdurahim, Al-Sharqi, F., Bahri, S. & Maharani, A.E.S.H. 2025b. On energy of prime coprime graph for dihedral groups. Mathematics and Statistics 13(3): 175-179.

Romdhini, M.U., Al-Sharqi, F., Nawawi, A., Al-Quran, A. & Rashmanlou, H. 2023. Signless Laplacian energy of interval-valued fuzzy graph and its applications. Sains Malaysiana 52(07): 2127-2138.

Saini, M., Kumari, V., Rana, P., Sehgal, A. & Singh, D. 2025. Degree based matrices of co-prime order graph of finite groups. Mathematics and Statistics 13(3): 127-135.

Sehgal, A., Manjeet & Singh, D. 2021. Co-prime order graphs of finite Abelian groups and dihedral groups. Journal of Mathematics and Computer Science 23: 196-202.

Tabraiz, A., Mufti, Z.S., Aslam, M.N., Saleem, N. & Zadeh, H. 2023. Fuzzy computational analysis of flower graph via fuzzy topological indices. Journal of Mathematics 2023: 8077729.

Wang, Y., Shahzadi, A., Binyamin, M.A., Mehtab, M., Tchier, F. & Aslam, A. 2024. On bounds for the atom bond sum connectivity index of graphs associated with symmetric numerical semigroups. AKCE International Journal of Graphs and Combinatorics 22(2): 125-135.

Zadeh, L.A. 1965. Fuzzy sets. Information and Control 8: 338-353.