Zagreb-Based Indices of Line Graph of Prime Coprime Graph for Integers Modulo Group
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Keywords:
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Line Graph, Prime Coprime Graph, Integers Modulo Group, Zagreb-Based Indices
Abstract
In this paper, we investigate the line graph of the prime coprime graph associated with the integers modulo group. Explicit general formulas are derived for the first Zagreb index, the second Zagreb index, and the hyper-Zagreb index of the considered structures. A comparative analysis is performed between the newly obtained results and previously reported findings, highlighting structural differences and index growth behaviour under the line-graph transformation. Furthermore, a statistical analysis is conducted to explore the quantitative relationship between the prime coprime graph and its corresponding line graph with respect to the computed Zagreb-based indices. The results provide deeper insight into the structural complexity of algebraically defined graphs and clarify how degree-based topological descriptors evolve under graph transformations.
References
Abdurahim, A., M. U. Romdhini, and S. K. S. Husain (2025). Zagreb-Based Indices of Prime Coprime Graph for Integers Modulo Power of Primes. Malaysian Journal of Mathematical Sciences, 19(3); 961–974
Abdurahim, A., M. U. Romdhini, J. Qudsi, F. Al-Sharqi, and Z. Md Rodzi (2026). Delta Degree-Based Indices of Prime Coprime Graph for Integers Modulo Group. Science and Technology Indonesia, 11(1); 10–18
Abel, L. A., D.Welyyanti, L. Yulianti, and D. Permana (2025). The Locating ChromaticNumber of the Cyclic Chain Graph. Science and Technology Indonesia, 10(3); 958–962
Adhikari, A. and S. S. Banerjee (2021). Prime Coprime Graph of a Finite Group. Novi Sad Journal ofMathematics, 52; 11151
Ahmed,W. E., M. F. Hanif, M. Hussain, and M. K. Siddiqui (2025). A Comprehensive Study on Topological Indices and Entropy Measures for Terbium Niobate Using Logarithmic Regression Models. Computational Biology and Chemistry, 119; 108558
Arockiaraj, M., D. Paul, J. Clement, S. Tigga, K. Jacob, and K. Balasubramanian (2023). Expected Value of First Zagreb Connection Index in Random Cyclooctatetraene Chain, Random Polyphenyls Chain, and Random Chain Network. SAR and QSAR in Environmental Research, 34; 569–589
Chang, S., J. Li, and Y. Zheng (2024). The Eigenvalue Multiplicity of Line Graphs. Linear Algebra and its Applications, 703; 47–62
Chen, X. (2023). General Sum-Connectivity Index of a Graph and Its Line Graph. Applied Mathematics and Computation, 443; 127779
Das, K. C., K. Xu, and J. Nam (2015). Zagreb Indices of Graphs. Frontiers of Mathematics in China, 10(3); 567–582
Durgut, R. and T. Turacı (2023). Computing Some Eccentricity-Based Topological Indices of Para-Line Graphs of Hexagonal Cactus Chains. Polycyclic Aromatic Compounds, 43(1); 115–130
Estrada, E. (2022). Statistical–Mechanical Theory of Topological Indices. Physica A: StatisticalMechanics and its Applications, 602; 127612
Fitriani, F., I. E.Wijayanti, A. Faisol, and S. Ali (2025). Commuting and Centralizing Maps on Modules. Science and Technology Indonesia, 10(3); 690–697
Gayathri, K. B. and S. Roy (2025). Quantitative Structure Property Relationship Study of Postpartum Depression Medications Using Topological Indices and RegressionModels. Ain Shams Engineering Journal, 16(1); 103194
Gonzalez-Diaz, H., S. Vilar, L. Santana, and E. Uriarte (2007). Medicinal Chemistry and Bioinformatics - Current Trends in Drugs Discovery with Networks Topological Indices. Current Topics in Medicinal Chemistry, 7(10); 569–589
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
Hakeem, A., A. Ullah, S. Zaman, E. E. Mahmoud, H. Ahmad, P. Ali, and M. B. Belay (2025). Topological Modeling and QSPR Based Prediction of Physicochemical Properties of Bioactive Polyphenols. Scientific Reports, 15(27466); 1–10
Hassan, M. M. (2023). Connection Number-based Multiplicative Zagreb Indices of Chemical Structures. Current Organic Chemistry, 27(21); 1873–1881
Islam, S. R., B. Mohsin, and M. Pal (2024). Hyper-Zagreb Index in Fuzzy Environment and Its Application. Heliyon, 10(16); e36110
Li, G. and M. Zhang (2022). Sharp Bounds on the Arithmetic-Geometric Index of Graphs and Line Graphs. Discrete Applied Mathematics, 318; 47–60
Mahboob, A., M. W. Rasheed, A. M. Dhiaa, I. Hanif, and L. Amin (2024). On Quantitative Structure-Property Relationship (QSPR) Analysis of Physicochemical Properties and Anti-Hepatitis Prescription Drugs Using a Linear Regression Model. Heliyon, 10(4); e25908
Mannan, M. A., H. Akter, and M. A. Ullah (2022). Evaluate All The Order of Every Element in The Higher Even, Odd, and Prime Order of Group for Composition. Science and Technology Indonesia, 7(3); 333–343
Manonmani, R. and P. Selvarani (2025). Multilinear Regression Analysis of Antiviral Medications with Topological Indices. Journal of Molecular Graphics and Modelling, 140; 109104
Noureen, S., A. Ali, A. A. Bhatti, A. M. Alanazi, and Y. Shang (2024). Predicting Enthalpy of Formation of Benzenoid Hydrocarbons and OrderingMolecular Trees Using General Multiplicative Zagreb Indices. Heliyon, 10(10); e30913
Pongpipat, D. and N. Nupo (2026). On Vertex-Degree-Based Topological Indices of Cayley Digraphs of Rectangular Groups and Their Role as Molecular Descriptors. Science and Technology Indonesia, 11(2); 732–741
Raza, Z., S. Akhter, and Y. Shang (2023). Expected Value of First Zagreb Connection Index in Random Cyclooctatetraene Chain, Random Polyphenyls Chain, and Random Chain Network. Frontiers in Chemistry, 10; 1067874
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
Rosary, M. S. (2022). Topological Study of Line Graph of Remdesivir Compound Used in the Treatment of Corona Virus. Polycyclic Aromatic Compounds, 42(8); 5731–5747
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
Ul Ain, H., R. Ismail, M. K. Jamil, M. E. M. Abdulla, and M. Azeem (2024). Extremal Graph of Super Line Graph Operation Via Generalized Randić Index. Ain Shams Engineering Journal, 15(10); 102938
Usman, M. and M. Javaid (2024). Connection-Based Zagreb Indices of Polycyclic Aromatic Hydrocarbons Structures. Current Organic Synthesis, 21(3); 246–256
Yow, K. S., M. A. Javed, A. Q. Baig, M. Ahmad, S. Ghareeb, and A. Qayyum (2026). Computational Analysis of Non-Kekule Benzenoid Molecules for Medical Applications. Malaysian Journal of Mathematical Sciences, 20(1); 199–211
Zhen, W., D. Wong, and S. Xu (2025). Line Graphs with the Largest Eigenvalue Multiplicity. Discrete Mathematics, 348(9); 114562
Abdurahim, A., M. U. Romdhini, J. Qudsi, F. Al-Sharqi, and Z. Md Rodzi (2026). Delta Degree-Based Indices of Prime Coprime Graph for Integers Modulo Group. Science and Technology Indonesia, 11(1); 10–18
Abel, L. A., D.Welyyanti, L. Yulianti, and D. Permana (2025). The Locating ChromaticNumber of the Cyclic Chain Graph. Science and Technology Indonesia, 10(3); 958–962
Adhikari, A. and S. S. Banerjee (2021). Prime Coprime Graph of a Finite Group. Novi Sad Journal ofMathematics, 52; 11151
Ahmed,W. E., M. F. Hanif, M. Hussain, and M. K. Siddiqui (2025). A Comprehensive Study on Topological Indices and Entropy Measures for Terbium Niobate Using Logarithmic Regression Models. Computational Biology and Chemistry, 119; 108558
Arockiaraj, M., D. Paul, J. Clement, S. Tigga, K. Jacob, and K. Balasubramanian (2023). Expected Value of First Zagreb Connection Index in Random Cyclooctatetraene Chain, Random Polyphenyls Chain, and Random Chain Network. SAR and QSAR in Environmental Research, 34; 569–589
Chang, S., J. Li, and Y. Zheng (2024). The Eigenvalue Multiplicity of Line Graphs. Linear Algebra and its Applications, 703; 47–62
Chen, X. (2023). General Sum-Connectivity Index of a Graph and Its Line Graph. Applied Mathematics and Computation, 443; 127779
Das, K. C., K. Xu, and J. Nam (2015). Zagreb Indices of Graphs. Frontiers of Mathematics in China, 10(3); 567–582
Durgut, R. and T. Turacı (2023). Computing Some Eccentricity-Based Topological Indices of Para-Line Graphs of Hexagonal Cactus Chains. Polycyclic Aromatic Compounds, 43(1); 115–130
Estrada, E. (2022). Statistical–Mechanical Theory of Topological Indices. Physica A: StatisticalMechanics and its Applications, 602; 127612
Fitriani, F., I. E.Wijayanti, A. Faisol, and S. Ali (2025). Commuting and Centralizing Maps on Modules. Science and Technology Indonesia, 10(3); 690–697
Gayathri, K. B. and S. Roy (2025). Quantitative Structure Property Relationship Study of Postpartum Depression Medications Using Topological Indices and RegressionModels. Ain Shams Engineering Journal, 16(1); 103194
Gonzalez-Diaz, H., S. Vilar, L. Santana, and E. Uriarte (2007). Medicinal Chemistry and Bioinformatics - Current Trends in Drugs Discovery with Networks Topological Indices. Current Topics in Medicinal Chemistry, 7(10); 569–589
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
Hakeem, A., A. Ullah, S. Zaman, E. E. Mahmoud, H. Ahmad, P. Ali, and M. B. Belay (2025). Topological Modeling and QSPR Based Prediction of Physicochemical Properties of Bioactive Polyphenols. Scientific Reports, 15(27466); 1–10
Hassan, M. M. (2023). Connection Number-based Multiplicative Zagreb Indices of Chemical Structures. Current Organic Chemistry, 27(21); 1873–1881
Islam, S. R., B. Mohsin, and M. Pal (2024). Hyper-Zagreb Index in Fuzzy Environment and Its Application. Heliyon, 10(16); e36110
Li, G. and M. Zhang (2022). Sharp Bounds on the Arithmetic-Geometric Index of Graphs and Line Graphs. Discrete Applied Mathematics, 318; 47–60
Mahboob, A., M. W. Rasheed, A. M. Dhiaa, I. Hanif, and L. Amin (2024). On Quantitative Structure-Property Relationship (QSPR) Analysis of Physicochemical Properties and Anti-Hepatitis Prescription Drugs Using a Linear Regression Model. Heliyon, 10(4); e25908
Mannan, M. A., H. Akter, and M. A. Ullah (2022). Evaluate All The Order of Every Element in The Higher Even, Odd, and Prime Order of Group for Composition. Science and Technology Indonesia, 7(3); 333–343
Manonmani, R. and P. Selvarani (2025). Multilinear Regression Analysis of Antiviral Medications with Topological Indices. Journal of Molecular Graphics and Modelling, 140; 109104
Noureen, S., A. Ali, A. A. Bhatti, A. M. Alanazi, and Y. Shang (2024). Predicting Enthalpy of Formation of Benzenoid Hydrocarbons and OrderingMolecular Trees Using General Multiplicative Zagreb Indices. Heliyon, 10(10); e30913
Pongpipat, D. and N. Nupo (2026). On Vertex-Degree-Based Topological Indices of Cayley Digraphs of Rectangular Groups and Their Role as Molecular Descriptors. Science and Technology Indonesia, 11(2); 732–741
Raza, Z., S. Akhter, and Y. Shang (2023). Expected Value of First Zagreb Connection Index in Random Cyclooctatetraene Chain, Random Polyphenyls Chain, and Random Chain Network. Frontiers in Chemistry, 10; 1067874
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
Rosary, M. S. (2022). Topological Study of Line Graph of Remdesivir Compound Used in the Treatment of Corona Virus. Polycyclic Aromatic Compounds, 42(8); 5731–5747
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
Ul Ain, H., R. Ismail, M. K. Jamil, M. E. M. Abdulla, and M. Azeem (2024). Extremal Graph of Super Line Graph Operation Via Generalized Randić Index. Ain Shams Engineering Journal, 15(10); 102938
Usman, M. and M. Javaid (2024). Connection-Based Zagreb Indices of Polycyclic Aromatic Hydrocarbons Structures. Current Organic Synthesis, 21(3); 246–256
Yow, K. S., M. A. Javed, A. Q. Baig, M. Ahmad, S. Ghareeb, and A. Qayyum (2026). Computational Analysis of Non-Kekule Benzenoid Molecules for Medical Applications. Malaysian Journal of Mathematical Sciences, 20(1); 199–211
Zhen, W., D. Wong, and S. Xu (2025). Line Graphs with the Largest Eigenvalue Multiplicity. Discrete Mathematics, 348(9); 114562
Authors
Abdurahim, Romdhini, M. U., & Qudsi, J. (2026). Zagreb-Based Indices of Line Graph of Prime Coprime Graph for Integers Modulo Group. Science and Technology Indonesia, 11(3), 844–854. https://doi.org/10.26554/sti.2026.11.3.844-854
This work is licensed under a Creative Commons Attribution 4.0 International License.