A Remark on the Energy Bounds and Degree-Based Topological Indices of the Zero-Divisor Graph ℤₚ[x]/⟨x⁴⟩
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Keywords:
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Graph Energy, Degree-Based Topological Indices, Zero-Divisor Graph
Abstract
The zero-divisor graph of a commutative ring, Γ(R), is a prominent subject in graph theory. Previous research has investigated the energy and topological indices of Γ(ℤₚ[x]/⟨x⁴⟩) for any prime p; however, errors were identified in the characteristic polynomial and the associated energy bounds. Recent work by Rather addressed these discrepancies using a vertex partitioning and quotient matrix approach. In contrast, this study presents a distinct matrix-theoretic approach based on the Schur complement to systematically derive the characteristic polynomial and its corresponding eigenvalues, thereby independently confirming and reinforcing the corrected energy bounds. Furthermore, this study significantly extends the existing literature by investigating a broader range of previously unexplored degree-based topological indices, including the second modified Zagreb, general and inverse general Randić, third and fifth symmetric division, harmonic, inverse sum, and forgotten indices.
References
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Ashraf, M., M. R. Mozumder, M. Rashid, and Nazim (2024). On a�� Spectrum of the Zero-Divisor Graph of the Ring ℤn. Discrete Mathematics, Algorithms and Applications, 16(04); 2350036
Bajaj, S. and P. Panigrahi (2022). On the Adjacency Spectrum of Zero Divisor Graph of Ring ℤn. Journal of Algebra and Its Applications, 21(10); 2250197
Baser, Z. and V. Ulucay (2025). Energy of a Neutrosophic Soft Set and Its Applications to Multi-Criteria Decision-Making Problems. Neutrosophic Sets and Systems, 79(1); 28
Beck, I. (1988). Coloring of Commutative Rings. Journal of Algebra, 116(1); 208–226
Biler, P. and A. Witkowski (2017). Problems in Mathematical Analysis. CRC Press
Bokhary, S. A. U. H., Adnan, M. K. Siddiqui, and M. Cancan (2022). On Topological Indices and QSPR Analysis of Drugs Used for the Treatment of Breast Cancer. Polycyclic Aromatic Compounds, 42(9); 6233–6253
Bollobás, B. and P. Erdös (1998). Graphs of Extremal Weights. Ars Combinatoria, 50; 225
Cruz, R., C. Espinal, and J. J. M. Rada (2024). A Matrix Approach to Vertex-Degree-Based Topological Indices. Mathematics, 12(13); 2043
Elahi, K., A. Ahmad, M. Asim, and R. Hasni (2022). Computation of Edge-Based Topological Indices for Zero Divisor Graphs of Commutative Rings. Italian Journal of Pure and Applied Mathematics, 48; 523–534
Furtula, B. and I. Gutman (2015). A Forgotten Topological Index. Journal of Mathematical Chemistry, 53(4); 1184–1190
Graovac, A., I. Gutman, and N. Trinajstić (1976). A Linear Relationship between the Total π-Electron Energy and the Characteristic Polynomial. Chemical Physics Letters, 37(3); 471–474
Gutman, I. (1978). The Energy of a Graph. Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz, 103; 1–22
Gutman, I. and N. Trinajstić (2005). New Concepts II. In Graph Theory and Molecular Orbitals, volume 17. Springer, pages 49–93
Jamal, F. and M. Imran (2024). Distance Spectrum of Some Zero Divisor Graphs. AIMS Mathematics, 9; 23979–23996
Jannat, F. E. and R. K. Nath (2026). On Distance Spectra, Energies and Wiener Index of Non-Commuting Conjugacy Class Graphs. Journal of Algebra Combinatorics Discrete Structures Applications; 43–73
Johnson, C. and R. Sankar (2023). Graph Energy and Topological Descriptors of Zero Divisor Graph Associated with Commutative Ring. Journal of Applied Mathematics and Computing, 69(3); 2641–2656
Koam, A. N., A. Ahmad, A. Haider, and M. A. Ansari (2022). Computation of Eccentric Topological Indices of Zero Divisor Graphs Based on Their Edges. AIMS Mathematics, 7(7); 11509–11518
Magi, P., S. M. Jose, and A. Kishore (2020). Adjacency Matrix and Eigenvalues of the Zero Divisor Graph Γ(ℤn). Journal of Mathematical and Computational Science, 10(4); 1285–1297
Miličević, A., S. Nikolić, and N. Trinajstić (2004). On Reformulated Zagreb Indices. Molecular Diversity, 8(4); 393–399
Mondal, S., M. Imran, N. De, and A. Pal (2023). Topological Indices of Total Graph and Zero Divisor Graph of Commutative Ring: A Polynomial Approach. Complexity, 2023(1); 6815657
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Oboudi, M. R. (2023). On Graphs with Integer Sombor Index. Journal of Applied Mathematics and Computing, 69(1); 941–952
Pirzada, S., B. A. Wani, and A. Somasundaram (2021). On the Eigenvalues of Zero-Divisor Graph Associated to Finite Commutative Ring. AKCE International Journal of Graphs and Combinatorics, 18(1); 1–6
Rather, B. A. (2026). On the Eigenvalues of Zero Divisor Graphs Associated with Commutative Rings. Indian Journal of Pure and Applied Mathematics; 1–16
Rather, B. A., F. Ali, A. Ullah, N. Fatima, and R. Dad (2022). a�� Eigenvalues of Zero Divisor Graph of Integer Modulo and Von Neumann Regular Rings. Symmetry, 14(8); 1710
Rather, B. A., S. Pirzada, T. A. Naikoo, and Y. Shang (2021). On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n. Mathematics, 9(5); 482
Rauf, A., M. Naeem, and S. U. Bukhari (2022). Quantitative Structure–Property Relationship of Ev-Degree and Ve-Degree Based Topological Indices: Physico-Chemical Properties of Benzene Derivatives. International Journal of Quantum Chemistry, 122(5); e26851
Rayer, C. and R. Jeyaraj (2023). Applications on Topological Indices of Zero-Divisor Graph Associated with Commutative Rings. Symmetry, 15(2); 335
Rehman, N. u., Nazim, and M. Nazim (2024). Exploring Normalized Distance Laplacian Eigenvalues of the Zero Divisor Graph of Ring ℤn. Rendiconti del Circolo Matematico di Palermo Series 2, 73(2); 515–526
Singh, P. and V. K. Bhat (2021). Adjacency Matrix and Wiener Index of Zero Divisor Graph Γ(ℤn). Journal of Applied Mathematics and Computing, 66(1); 717–732
Vukičević, D. and M. Gašperov (2010). Bond Additive Modeling 1. Adriatic Indices. Croatica Chemica Acta, 83(3); 243–260
Wang, Z., Y. Mao, Y. Li, and B. Furtula (2022). On Relations between Sombor and Other Degree-Based Indices. Journal of Applied Mathematics and Computing, 68(1); 1–17
Zhang, F. (2006). The Schur Complement and Its Applications. Springer Science & Business Media
Zhang, K., Y. Liu, L. Zhang, C. Ma, and Y. Xu (2025). Frequency Slice Graph Spectrum Model and Its Application in Bearing Fault Feature Extraction. Mechanical Systems and Signal Processing, 226; 112383
Zhong, L. (2012). The Harmonic Index for Graphs. Applied Mathematics Letters, 25(3); 561–566
Ahmad, A. and S. López (2021). Distance-Based Topological Polynomials Associated with Zero-Divisor Graphs. Mathematical Problems in Engineering, 2021(1); 4959559
Ahmadi, M. R. and R. Jahani-Nezhad (2011). Energy and Wiener Index of Zero-Divisor Graphs. Iranian Journal of Mathematical Chemistry, 2; 45–51
Alali, A. S., S. Ali, N. Hassan, A. M. Mahnashi, Y. Shang, and A. Assiry (2023). Algebraic Structure Graphs Over the Commutative Ring ℤm: Exploring Topological Indices and Entropies Using M-Polynomials. Mathematics, 11(18); 3833
Anderson, D. F. and S. B. Mulay (2007). On the Diameter and Girth of a Zero-Divisor Graph. Journal of Pure and Applied Algebra, 210(2); 543–550
Ashraf, M., M. R. Mozumder, M. Rashid, and Nazim (2024). On a�� Spectrum of the Zero-Divisor Graph of the Ring ℤn. Discrete Mathematics, Algorithms and Applications, 16(04); 2350036
Bajaj, S. and P. Panigrahi (2022). On the Adjacency Spectrum of Zero Divisor Graph of Ring ℤn. Journal of Algebra and Its Applications, 21(10); 2250197
Baser, Z. and V. Ulucay (2025). Energy of a Neutrosophic Soft Set and Its Applications to Multi-Criteria Decision-Making Problems. Neutrosophic Sets and Systems, 79(1); 28
Beck, I. (1988). Coloring of Commutative Rings. Journal of Algebra, 116(1); 208–226
Biler, P. and A. Witkowski (2017). Problems in Mathematical Analysis. CRC Press
Bokhary, S. A. U. H., Adnan, M. K. Siddiqui, and M. Cancan (2022). On Topological Indices and QSPR Analysis of Drugs Used for the Treatment of Breast Cancer. Polycyclic Aromatic Compounds, 42(9); 6233–6253
Bollobás, B. and P. Erdös (1998). Graphs of Extremal Weights. Ars Combinatoria, 50; 225
Cruz, R., C. Espinal, and J. J. M. Rada (2024). A Matrix Approach to Vertex-Degree-Based Topological Indices. Mathematics, 12(13); 2043
Elahi, K., A. Ahmad, M. Asim, and R. Hasni (2022). Computation of Edge-Based Topological Indices for Zero Divisor Graphs of Commutative Rings. Italian Journal of Pure and Applied Mathematics, 48; 523–534
Furtula, B. and I. Gutman (2015). A Forgotten Topological Index. Journal of Mathematical Chemistry, 53(4); 1184–1190
Graovac, A., I. Gutman, and N. Trinajstić (1976). A Linear Relationship between the Total π-Electron Energy and the Characteristic Polynomial. Chemical Physics Letters, 37(3); 471–474
Gutman, I. (1978). The Energy of a Graph. Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz, 103; 1–22
Gutman, I. and N. Trinajstić (2005). New Concepts II. In Graph Theory and Molecular Orbitals, volume 17. Springer, pages 49–93
Jamal, F. and M. Imran (2024). Distance Spectrum of Some Zero Divisor Graphs. AIMS Mathematics, 9; 23979–23996
Jannat, F. E. and R. K. Nath (2026). On Distance Spectra, Energies and Wiener Index of Non-Commuting Conjugacy Class Graphs. Journal of Algebra Combinatorics Discrete Structures Applications; 43–73
Johnson, C. and R. Sankar (2023). Graph Energy and Topological Descriptors of Zero Divisor Graph Associated with Commutative Ring. Journal of Applied Mathematics and Computing, 69(3); 2641–2656
Koam, A. N., A. Ahmad, A. Haider, and M. A. Ansari (2022). Computation of Eccentric Topological Indices of Zero Divisor Graphs Based on Their Edges. AIMS Mathematics, 7(7); 11509–11518
Magi, P., S. M. Jose, and A. Kishore (2020). Adjacency Matrix and Eigenvalues of the Zero Divisor Graph Γ(ℤn). Journal of Mathematical and Computational Science, 10(4); 1285–1297
Miličević, A., S. Nikolić, and N. Trinajstić (2004). On Reformulated Zagreb Indices. Molecular Diversity, 8(4); 393–399
Mondal, S., M. Imran, N. De, and A. Pal (2023). Topological Indices of Total Graph and Zero Divisor Graph of Commutative Ring: A Polynomial Approach. Complexity, 2023(1); 6815657
Nalina, C., P. S. K. Reddy, M. Kirankumar, and M. Pavithra (2026). On Adjacency Square Root Stress Sum Eigenvalues and Energy of Graphs. Proceedings of the Jangjeon Mathematical Society, 29(1); 51–64
Narumi, H. and M. Katayama (1984). Simple Topological Index: A Newly Devised Index Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons. Memoirs of the Faculty of Engineering, Hokkaido University, 16(3); 209–214
Oboudi, M. R. (2023). On Graphs with Integer Sombor Index. Journal of Applied Mathematics and Computing, 69(1); 941–952
Pirzada, S., B. A. Wani, and A. Somasundaram (2021). On the Eigenvalues of Zero-Divisor Graph Associated to Finite Commutative Ring. AKCE International Journal of Graphs and Combinatorics, 18(1); 1–6
Rather, B. A. (2026). On the Eigenvalues of Zero Divisor Graphs Associated with Commutative Rings. Indian Journal of Pure and Applied Mathematics; 1–16
Rather, B. A., F. Ali, A. Ullah, N. Fatima, and R. Dad (2022). a�� Eigenvalues of Zero Divisor Graph of Integer Modulo and Von Neumann Regular Rings. Symmetry, 14(8); 1710
Rather, B. A., S. Pirzada, T. A. Naikoo, and Y. Shang (2021). On Laplacian Eigenvalues of the Zero-Divisor Graph Associated to the Ring of Integers Modulo n. Mathematics, 9(5); 482
Rauf, A., M. Naeem, and S. U. Bukhari (2022). Quantitative Structure–Property Relationship of Ev-Degree and Ve-Degree Based Topological Indices: Physico-Chemical Properties of Benzene Derivatives. International Journal of Quantum Chemistry, 122(5); e26851
Rayer, C. and R. Jeyaraj (2023). Applications on Topological Indices of Zero-Divisor Graph Associated with Commutative Rings. Symmetry, 15(2); 335
Rehman, N. u., Nazim, and M. Nazim (2024). Exploring Normalized Distance Laplacian Eigenvalues of the Zero Divisor Graph of Ring ℤn. Rendiconti del Circolo Matematico di Palermo Series 2, 73(2); 515–526
Singh, P. and V. K. Bhat (2021). Adjacency Matrix and Wiener Index of Zero Divisor Graph Γ(ℤn). Journal of Applied Mathematics and Computing, 66(1); 717–732
Vukičević, D. and M. Gašperov (2010). Bond Additive Modeling 1. Adriatic Indices. Croatica Chemica Acta, 83(3); 243–260
Wang, Z., Y. Mao, Y. Li, and B. Furtula (2022). On Relations between Sombor and Other Degree-Based Indices. Journal of Applied Mathematics and Computing, 68(1); 1–17
Zhang, F. (2006). The Schur Complement and Its Applications. Springer Science & Business Media
Zhang, K., Y. Liu, L. Zhang, C. Ma, and Y. Xu (2025). Frequency Slice Graph Spectrum Model and Its Application in Bearing Fault Feature Extraction. Mechanical Systems and Signal Processing, 226; 112383
Zhong, L. (2012). The Harmonic Index for Graphs. Applied Mathematics Letters, 25(3); 561–566
Authors
Krisnawati, V. H. ., Musyarrofah, A. F., & Andari, A. (2026). A Remark on the Energy Bounds and Degree-Based Topological Indices of the Zero-Divisor Graph ℤₚ[x]/⟨x⁴⟩. Science and Technology Indonesia, 11(4), 1596–1607. https://doi.org/10.26554/sti.2026.11.4.1596-1607
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