Delta Degree-Based Indices of Prime Coprime Graph for Integers Modulo Group
Article Sidebar
Keywords:
-
Prime Coprime Graph, Delta Degree, Topological Indices, Integers Modulo Group
Abstract
Research on prime coprime graphs of finite groups has largely focused on structural properties, spectra, and classical topological indices, with limited attention given to delta degree-based indices. To address this gap, this study investigates delta degree-based topological indices of the prime coprime graph constructed on the group of integers modulo n, Zn. In this graph, the vertices correspond to the elements of Zn, and two distinct vertices are adjacent if and only if the greatest common divisor of their orders is either 1 or a prime number. In the present work, the focus lies on computing and analyzing several delta degree-based topological indices that are obtained by incorporating the concept of delta degree into classical topological indices, including the delta first Zagreb index, the delta second Zagreb index, the delta hyper Zagreb index, and the delta forgotten index. The methodology involves deriving formulas for these delta-based indices for various values of n, supported by systematic computations and data tabulation. Beyond purely algebraic computation, statistical tools are employed to investigate the relationships between different indices. In particular, a comparative distribution analysis is conducted to determine whether pairs of indices exhibit similar patterns of variability using the Levene test.
References
Abdurahim, A., M. U. Romdhini, F. Al-Sharqi, and N. A. Robbaniyah (2025a). Relative Prime Coprime Graph of Integers Modulo Group and Its Reverse Topological Indices. Pan American Journal of Mathematics, 4(10); 1–9
Abdurahim, A., M. U. Romdhini, and S. K. S. Husain (2025b). Zagreb-Based Indices of Prime Coprime Graph for Integers Modulo Power of Primes. Malaysian Journal of Mathematical Sciences, 19(3); 961–974
Adhikari, A. and S. Banerjee (2021). Prime Coprime Graph of a Finite Group. Novi Sad Journal of Mathematics, 52; 11151
Ahmad, A., A. N. Koam, and M. Azeem (2023). Reverse-Degree-Based Topological Indices of Fullerene Cage Networks. Molecular Physics, 121(14); e2212533
Ahmad, M., Y. Yusof, A. Qayyum, L. Rathour, V. Singh, and L. N. Mishra (2025). A New Development in Topological Analysis of Propane Para-Line Graphs With Application in Chemical Composition. Discontinuity, Nonlinearity, and Complexity, 14(4); 659–668
Arif, W., W. A. Khan, A. Khan, and H. Rashmanlou (2023). Some Indices of Picture Fuzzy Graphs and Their Applications. Computational and Applied Mathematics, 42; 253
Asghar, A. (2025). Nm-Polynomial and Neighborhood Degree-Based Indices in Graph Theory: A Study on Non-Kekulean Benzenoid Graphs. Acta Chimica Asiana, 8(1); 555–563
Das, K. C., K. Xu, and J. Nam (2015). Zagreb Indices of Graphs. Frontiers of Mathematics in China, 10(3); 567–582
Dummit, D. S. and R. M. Foote (2004). Abstract Algebra. Wiley, Hoboken, 3 edition
Ediz, S. and M. Cancan (2016). Reverse Zagreb Indices of Cartesian Product of Graphs. International Journal of Mathematics and Computer Science, 11(1); 51–58
Fitriani, I. E. Wijayanti, A. Faisol, and S. Ali (2025). Commuting and Centralizing Maps on Modules. Science and Technology Indonesia, 10(3); 690–697
Fujita, T. and T. Smarandache (2025a). General, General Weak, Anti, Balanced, and Semi-Neutrosophic Graph. Neutrosophic Sets and Systems, 85; 398–435
Fujita, T. and T. Smarandache (2025b). Neutrosophic Soft N-Super-Hypergraphs With Real-World Applications. European Journal of Pure and Applied Mathematics, 18(3); 6621
Fujita, T. and T. Smarandache (2025c). Soft Directed N-Superhypergraphs With Some Real-World Applications. European Journal of Pure and Applied Mathematics, 18(4); 1–29
Gao, W., M. Younas, A. Farooq, A. U. R. Virk, and W. Nazeer (2018). Some Reverse Degree-Based Topological Indices and Polynomials of Dendrimers. Mathematics, 6(10); 214
Gowtham, K. J. and M. N. Husin (2023a). A Study of Families of Bistar and Corona Product of Graph: Reverse Topological Indices. Malaysian Journal of Mathematical Sciences, 17(4); 575–586
Gowtham, K. J. and M. N. Husin (2023b). A Study of Reverse Topological Indices and Their Importance in Chemical Sciences. Applied Mathematics E-Notes, 23; 175–186
Gunderson, D. S. and K. H. Rosen (2010). Handbook of Mathematical Induction. CRC Press, Boca Raton
Gutman, I. (2021). Geometric Approach to Degree-Based Topological Indices: Sombor Indices. MATCH Communications in Mathematical and in Computer Chemistry, 86(1); 11–16
Gutman, I. and K. C. Das (2004). The First Zagreb Index 30 Years After. MATCH Communications in Mathematical and in Computer Chemistry, 50; 83–92
Gutman, I. and O. E. Polansky (2012). Mathematical Concepts in Organic Chemistry. Springer Science & Business Media
Kulli, V. R. (2018). Multiplicative Connectivity Indices of Nanostructures. LAP Lambert Academic Publishing, Riga
Kulli, V. R. (2021a). -Banhatti, Hyper -Banhatti Indices and Their Polynomials of Certain Networks. International Journal of Engineering Sciences and Research Technology, 10(3); 1–8
Kulli, V. R. (2021b). -Sombor Index and Its Exponential for Certain Nanotubes. Annals of Pure and Applied Mathematics, 23(1); 37–42
Kulli, V. R., J. A. Mendez-Bermudez, J. M. Rodríguez García, and J. M. Sigarreta Almira (2023). Revan Sombor Indices: Analytical and Statistical Study. Mathematical Biosciences and Engineering
Kwun, Y. C., A. U. R. Virk, M. Rafaqat, M. U. Rehman, and W. Nazeer (2019). Some Reversed Degree-Based Topological Indices for Graphene. Journal of Discrete Mathematical Sciences and Cryptography, 22(7); 1305–1314
Romdhini, M. U., A. Abdurahim, A. E. S. H. Maharani, and S. R. Kamali (2025). Transmission-Based Energies of Prime Coprime Graph for Integers Modulo Group. Science and Technology Indonesia, 10(3); 779–785
Romdhini, M. U. and A. Nawawi (2025). Relation Between the First Zagreb and Greatest Common Divisor Degree Energies of Commuting Graph for Dihedral Groups. Science and Technology Indonesia, 10(1); 1–8
Romdhini, M. U., A. Nawawi, F. Al-Sharqi, A. Al-Quran, and S. R. Kamali (2024). Wiener-Hosoya Energy of Non-Commuting Graph for Dihedral Groups. Asia Pacific Journal of Mathematics, 11(9); 1–9
Rosary, M. S. (2023). On Reverse Valency Based Topological Indices of Metal-Organic Framework. Polycyclic Aromatic Compounds, 43(1); 860–873
Rosen, K. H. (2011). Elementary Number Theory. Pearson Education, London
Saini, M., S. M. S. Khasraw, A. Sehgal, and D. Singh (2021). On Co-Prime Order Graphs of Finite Abelian P-Groups. Journal of Mathematical and Computational Science, 1(6); 7052–7061
Saini, M., V. Kumari, P. Rana, A. Sehgal, and D. Singh (2025). Degree Based Matrices of Co-Prime Order Graph of Finite Groups. Mathematics and Statistics, 13(3); 127–135
Saini, M., G. Singh, A. Sehgal, and D. Singh (2024). On Divisor Labeling of Co-Prime Order Graphs of Finite Groups. Italian Journal of Pure and Applied Mathematics, 51(3); 443–451
Shirdel, G. H., H. Rezapour, and A. M. Sayadi (2013). The Hyper-Zagreb Index of Graph Operations. Iranian Journal of Mathematical Chemistry, 4(2); 213–220
Wei, J., A. Khalid, P. Ali, M. K. Siddiqui, A. Nawaz, and M. Hussain (2023). Computing Reverse Degree Based Topological Indices of Vanadium Carbide. Polycyclic Aromatic Compounds, 43(2); 1172–1191
Abdurahim, A., M. U. Romdhini, and S. K. S. Husain (2025b). Zagreb-Based Indices of Prime Coprime Graph for Integers Modulo Power of Primes. Malaysian Journal of Mathematical Sciences, 19(3); 961–974
Adhikari, A. and S. Banerjee (2021). Prime Coprime Graph of a Finite Group. Novi Sad Journal of Mathematics, 52; 11151
Ahmad, A., A. N. Koam, and M. Azeem (2023). Reverse-Degree-Based Topological Indices of Fullerene Cage Networks. Molecular Physics, 121(14); e2212533
Ahmad, M., Y. Yusof, A. Qayyum, L. Rathour, V. Singh, and L. N. Mishra (2025). A New Development in Topological Analysis of Propane Para-Line Graphs With Application in Chemical Composition. Discontinuity, Nonlinearity, and Complexity, 14(4); 659–668
Arif, W., W. A. Khan, A. Khan, and H. Rashmanlou (2023). Some Indices of Picture Fuzzy Graphs and Their Applications. Computational and Applied Mathematics, 42; 253
Asghar, A. (2025). Nm-Polynomial and Neighborhood Degree-Based Indices in Graph Theory: A Study on Non-Kekulean Benzenoid Graphs. Acta Chimica Asiana, 8(1); 555–563
Das, K. C., K. Xu, and J. Nam (2015). Zagreb Indices of Graphs. Frontiers of Mathematics in China, 10(3); 567–582
Dummit, D. S. and R. M. Foote (2004). Abstract Algebra. Wiley, Hoboken, 3 edition
Ediz, S. and M. Cancan (2016). Reverse Zagreb Indices of Cartesian Product of Graphs. International Journal of Mathematics and Computer Science, 11(1); 51–58
Fitriani, I. E. Wijayanti, A. Faisol, and S. Ali (2025). Commuting and Centralizing Maps on Modules. Science and Technology Indonesia, 10(3); 690–697
Fujita, T. and T. Smarandache (2025a). General, General Weak, Anti, Balanced, and Semi-Neutrosophic Graph. Neutrosophic Sets and Systems, 85; 398–435
Fujita, T. and T. Smarandache (2025b). Neutrosophic Soft N-Super-Hypergraphs With Real-World Applications. European Journal of Pure and Applied Mathematics, 18(3); 6621
Fujita, T. and T. Smarandache (2025c). Soft Directed N-Superhypergraphs With Some Real-World Applications. European Journal of Pure and Applied Mathematics, 18(4); 1–29
Gao, W., M. Younas, A. Farooq, A. U. R. Virk, and W. Nazeer (2018). Some Reverse Degree-Based Topological Indices and Polynomials of Dendrimers. Mathematics, 6(10); 214
Gowtham, K. J. and M. N. Husin (2023a). A Study of Families of Bistar and Corona Product of Graph: Reverse Topological Indices. Malaysian Journal of Mathematical Sciences, 17(4); 575–586
Gowtham, K. J. and M. N. Husin (2023b). A Study of Reverse Topological Indices and Their Importance in Chemical Sciences. Applied Mathematics E-Notes, 23; 175–186
Gunderson, D. S. and K. H. Rosen (2010). Handbook of Mathematical Induction. CRC Press, Boca Raton
Gutman, I. (2021). Geometric Approach to Degree-Based Topological Indices: Sombor Indices. MATCH Communications in Mathematical and in Computer Chemistry, 86(1); 11–16
Gutman, I. and K. C. Das (2004). The First Zagreb Index 30 Years After. MATCH Communications in Mathematical and in Computer Chemistry, 50; 83–92
Gutman, I. and O. E. Polansky (2012). Mathematical Concepts in Organic Chemistry. Springer Science & Business Media
Kulli, V. R. (2018). Multiplicative Connectivity Indices of Nanostructures. LAP Lambert Academic Publishing, Riga
Kulli, V. R. (2021a). -Banhatti, Hyper -Banhatti Indices and Their Polynomials of Certain Networks. International Journal of Engineering Sciences and Research Technology, 10(3); 1–8
Kulli, V. R. (2021b). -Sombor Index and Its Exponential for Certain Nanotubes. Annals of Pure and Applied Mathematics, 23(1); 37–42
Kulli, V. R., J. A. Mendez-Bermudez, J. M. Rodríguez García, and J. M. Sigarreta Almira (2023). Revan Sombor Indices: Analytical and Statistical Study. Mathematical Biosciences and Engineering
Kwun, Y. C., A. U. R. Virk, M. Rafaqat, M. U. Rehman, and W. Nazeer (2019). Some Reversed Degree-Based Topological Indices for Graphene. Journal of Discrete Mathematical Sciences and Cryptography, 22(7); 1305–1314
Romdhini, M. U., A. Abdurahim, A. E. S. H. Maharani, and S. R. Kamali (2025). Transmission-Based Energies of Prime Coprime Graph for Integers Modulo Group. Science and Technology Indonesia, 10(3); 779–785
Romdhini, M. U. and A. Nawawi (2025). Relation Between the First Zagreb and Greatest Common Divisor Degree Energies of Commuting Graph for Dihedral Groups. Science and Technology Indonesia, 10(1); 1–8
Romdhini, M. U., A. Nawawi, F. Al-Sharqi, A. Al-Quran, and S. R. Kamali (2024). Wiener-Hosoya Energy of Non-Commuting Graph for Dihedral Groups. Asia Pacific Journal of Mathematics, 11(9); 1–9
Rosary, M. S. (2023). On Reverse Valency Based Topological Indices of Metal-Organic Framework. Polycyclic Aromatic Compounds, 43(1); 860–873
Rosen, K. H. (2011). Elementary Number Theory. Pearson Education, London
Saini, M., S. M. S. Khasraw, A. Sehgal, and D. Singh (2021). On Co-Prime Order Graphs of Finite Abelian P-Groups. Journal of Mathematical and Computational Science, 1(6); 7052–7061
Saini, M., V. Kumari, P. Rana, A. Sehgal, and D. Singh (2025). Degree Based Matrices of Co-Prime Order Graph of Finite Groups. Mathematics and Statistics, 13(3); 127–135
Saini, M., G. Singh, A. Sehgal, and D. Singh (2024). On Divisor Labeling of Co-Prime Order Graphs of Finite Groups. Italian Journal of Pure and Applied Mathematics, 51(3); 443–451
Shirdel, G. H., H. Rezapour, and A. M. Sayadi (2013). The Hyper-Zagreb Index of Graph Operations. Iranian Journal of Mathematical Chemistry, 4(2); 213–220
Wei, J., A. Khalid, P. Ali, M. K. Siddiqui, A. Nawaz, and M. Hussain (2023). Computing Reverse Degree Based Topological Indices of Vanadium Carbide. Polycyclic Aromatic Compounds, 43(2); 1172–1191
Authors
Abdurahim, Romdhini, M. U. ., Qudsi, J. ., Al-Sharqi, F. ., & Rodzi, Z. M. . (2026). Delta Degree-Based Indices of Prime Coprime Graph for Integers Modulo Group. Science and Technology Indonesia, 11(1), 10–18. https://doi.org/10.26554/sti.2026.11.1.10-18
This work is licensed under a Creative Commons Attribution 4.0 International License.