Main Article Content
Abstract
Topological indices are numerical graph invariants that reflect structural properties of graphs and have broad applications in chemistry, algebra, and network analysis. This paper focuses on the analysis of several topological indices in the context of unit graphs associated with modular integer rings. In a unit graph, vertices represent ring elements, and two vertices are adjacent if their sum is a unit. We investigate and derive general formulas for six indices: the Narumi-Katayama index, the Forgotten index, the Atom-Bond Connectivity (ABC) index, the first and second Gourava indices, and the first Revan index. Two cases are considered for the ring of integers modulo $n$, namely when $n$ is a power of $2$ and when $n$ is an odd prime. The results offer a deeper understanding of the algebraic and combinatorial properties of unit graphs and contribute to the development of algebraic graph theory.
Keywords
Article Details
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
All articles published in the Eigen Mathematics Journal will be available for free reading and downloading. The license applied to this journal is Creative Commons Attribution-Non-Commercial-Share Alike (CC BY-NC-SA).
References
- L. Euler, The Solution of a Problem Relating to the Geometry of Position. Comment. Acad. Sci. Petropolitanae, 1736. [Online]. Available: https://scholarlycommons.pacific.edu/euler-works/53.
- A. Cayley, On the Theory of the Analytical Forms called Trees, p. 242–246. Cambridge University Press, 2009.
- S. T. Lestari, J. R. Albaracin, and I. G. A. W. Wardhana, “Certain indexes of unit graph in integer modulo rings with specific orders,†BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 19, no. 4, pp. 2455–2466, 2025. https://doi.org/10.30598/barekengvol19iss4pp2455-2466.
- S. T. Lestari, P. K. Dewi, I. G. A. W. Wardhana, and I. N. Suparta, “Algebraic structures and combinatorial properties of unit graphs in rings of integer modulo with specific orders,†EIGEN MATHEMATICS JOURNAL, vol. 7, no. 2, pp. 89–92, 2024. https://doi.org/10.29303/emj.v7i2.235.
- E. Y. Asmarani, S. T. Lestari, D. Purnamasari, A. G. Syarifudin, S. Salwa, and I. Wardhana, “The first zagreb index, the wiener index, and the gutman index of the power of dihedral group,†CAUCHY: Jurnal Matematika Murni dan Aplikasi, vol. 7, no. 4, pp. 513–520, 2023. https://doi.org/10.18860/ca.v7i4.16991.
- M. N. Husni, H. Syafitri, A. M. Siboro, A. G. Syarifudin, Q. Aini, and I. G. A. W. Wardhana, “The harmonic index and the gutman index of coprime graph of integer group modulo with order of prime power,†BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 16, no. 3, pp. 961–966, 2022. https://doi.org/10.30598/barekengvol16iss3pp961-966.
- D. P. Malik, M. N. Husni, M. Miftahurrahman, I. G. A. Wardhana, and G. S. . Ismail, “The chemical topological graph associated with the nilpotent graph of a modulo ring of prime power order,†Journal of Fundamental Mathematics and Applications (JFMA), vol. 7, no. 1, pp. 1–9, 2024. https://doi.org/10.14710/jfma.v0i0.20269.
- M. Ascioglu and I. N. C. and, “Narumi–katayama index of the subdivision graphs,†Journal of Taibah University for Science, vol. 12, no. 4, pp. 401–408, 2018. https://doi.org/10.1080/16583655.2018.1474542.
- M. N. Jauhari, S. L. Jannah, T. Turmudi, and I. Nisfulaila, “On the forgotten index and Jacobson graphs associated with integer rings modulo n,†KUBIK: Jurnal Publikasi Ilmiah Matematika, vol. 9, p. 231–242, November 2024. https://doi.org/10.15575/kubik.v9i2.39585.
- “On abc eigenvalues and abc energy,†Linear Algebra and its Applications, vol. 544, pp. 141–157, 2018. https://doi.org/10.1016/j.laa.2018.01.011.
- V. Kulli, “The gourava indices and coindices of graphs,†Annals of Pure and Applied Mathematics, vol. 14, no. 1, pp. 33–38, 2017. http://dx.doi.org/10.22457/apam.v14n1a4.
- V. Kulli, “Revan indices of oxide and honeycomb networks,†International Journal of Mathematics and its Applications, vol. 55, p. 7, 2017. https://www.researchgate.net/publication/321385336 Revan indices of oxide and honeycomb networks.
References
L. Euler, The Solution of a Problem Relating to the Geometry of Position. Comment. Acad. Sci. Petropolitanae, 1736. [Online]. Available: https://scholarlycommons.pacific.edu/euler-works/53.
A. Cayley, On the Theory of the Analytical Forms called Trees, p. 242–246. Cambridge University Press, 2009.
S. T. Lestari, J. R. Albaracin, and I. G. A. W. Wardhana, “Certain indexes of unit graph in integer modulo rings with specific orders,†BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 19, no. 4, pp. 2455–2466, 2025. https://doi.org/10.30598/barekengvol19iss4pp2455-2466.
S. T. Lestari, P. K. Dewi, I. G. A. W. Wardhana, and I. N. Suparta, “Algebraic structures and combinatorial properties of unit graphs in rings of integer modulo with specific orders,†EIGEN MATHEMATICS JOURNAL, vol. 7, no. 2, pp. 89–92, 2024. https://doi.org/10.29303/emj.v7i2.235.
E. Y. Asmarani, S. T. Lestari, D. Purnamasari, A. G. Syarifudin, S. Salwa, and I. Wardhana, “The first zagreb index, the wiener index, and the gutman index of the power of dihedral group,†CAUCHY: Jurnal Matematika Murni dan Aplikasi, vol. 7, no. 4, pp. 513–520, 2023. https://doi.org/10.18860/ca.v7i4.16991.
M. N. Husni, H. Syafitri, A. M. Siboro, A. G. Syarifudin, Q. Aini, and I. G. A. W. Wardhana, “The harmonic index and the gutman index of coprime graph of integer group modulo with order of prime power,†BAREKENG: Jurnal Ilmu Matematika dan Terapan, vol. 16, no. 3, pp. 961–966, 2022. https://doi.org/10.30598/barekengvol16iss3pp961-966.
D. P. Malik, M. N. Husni, M. Miftahurrahman, I. G. A. Wardhana, and G. S. . Ismail, “The chemical topological graph associated with the nilpotent graph of a modulo ring of prime power order,†Journal of Fundamental Mathematics and Applications (JFMA), vol. 7, no. 1, pp. 1–9, 2024. https://doi.org/10.14710/jfma.v0i0.20269.
M. Ascioglu and I. N. C. and, “Narumi–katayama index of the subdivision graphs,†Journal of Taibah University for Science, vol. 12, no. 4, pp. 401–408, 2018. https://doi.org/10.1080/16583655.2018.1474542.
M. N. Jauhari, S. L. Jannah, T. Turmudi, and I. Nisfulaila, “On the forgotten index and Jacobson graphs associated with integer rings modulo n,†KUBIK: Jurnal Publikasi Ilmiah Matematika, vol. 9, p. 231–242, November 2024. https://doi.org/10.15575/kubik.v9i2.39585.
“On abc eigenvalues and abc energy,†Linear Algebra and its Applications, vol. 544, pp. 141–157, 2018. https://doi.org/10.1016/j.laa.2018.01.011.
V. Kulli, “The gourava indices and coindices of graphs,†Annals of Pure and Applied Mathematics, vol. 14, no. 1, pp. 33–38, 2017. http://dx.doi.org/10.22457/apam.v14n1a4.
V. Kulli, “Revan indices of oxide and honeycomb networks,†International Journal of Mathematics and its Applications, vol. 55, p. 7, 2017. https://www.researchgate.net/publication/321385336 Revan indices of oxide and honeycomb networks.