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Abstract
A topological index is a numerical value that provides information about the structure of a graph. Among various degree-based topological indices, the forgotten topological index (F-index) is of particular interest in this study. The F-index is calculated for the zero divisor graph of a ring R. In graph theory, the zero divisor graph of R is defined as a graph with vertex set the zero-divisors of R, and for distinct vertices a and b are adjacent if a · b = 0. This research focuses on the zero divisor graph of the commutative ring of integers modulo 2ρn where ρ is an odd prime and n is a positive integer. The objectives are to determine the set of all zero divisors, analyze the vertex degrees of the graph, and then compute the F-index of the zero divisor graph. Using algebraic techniques, we derive the degree of each vertex, the distribution of vertex degrees, and the number of edges in the graph. The general expression for the F-index of the zero divisor graph for the ring is established. The results contribute to understanding topological indices for algebraic structures, with potential applications in chemical graph theory and related disciplines.
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References
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References
K. Elahi, A. Ahmad, and R. Hasni, “Construction algorithm for zero divisor graphs of finite commutative rings and their vertex-based eccentric topological indices,” Mathematics, vol. 6, no. 12, p. 301, 2018. https://doi.org/10.3390/math6120301.
J. B. Fraleigh, A first course in abstract algebra. Pearson Education India, 2003.
D. Anderson and P. Livingston, “The zero-divisor graph of a commutative ring,” Journal of Algebra, vol. 217, pp. 434–447, 1999. https://doi.org/10.1006/jabr.1998.7840.
D. S. Gunderson and K. H. Rosen, Handbook of mathematical induction: Theory and applications. CRC Press, 2014.
D. F. Anderson and D. Weber, “The zero-divisor graph of a commutative ring without identity,” International Electronic Journal of Algebra, vol. 23, no. 23, pp. 176–202, 2018.
A. Cherrabi, H. Essannouni, E. Jabbouri, and A. Ouadfel, “On a new extension of the zero-divisor graph,” in Algebra Colloquium, vol. 27, pp. 469–476, World Scientific, 2020. https://doi.org/10.1142/S1005386720000383.
S. Bajaj and P. Panigrahi, “On the adjacency spectrum of zero divisor graph of ring Zn,” Journal of Algebra and Its Applications, vol. 21, no. 10, p. 2250197, 2022. https://doi.org/10.1142/S0219498822501973.
M. F. Hanif, H. Mahmood, and S. Ahmad, “On degree-based entropy measure for zero-divisor graphs,” Discrete Mathematics, Algorithms and Applications, vol. 16, no. 08, p. 2350104, 2024. https://doi.org/10.1142/S1793830923501045.
S. Mondal, N. De, and A. Pal, “Topological indices of some chemical structures applied for the treatment of covid-19 patients,” Polycyclic Aromatic Compounds, vol. 42, no. 4, pp. 1220–1234, 2022. https://doi.org/10.1080/10406638.2020.1770306.
B. Furtula and I. Gutman, “A forgotten topological index,” Journal of Mathematical Chemistry, vol. 53, pp. 1184–1190, 2015. https://doi.org/10.1007/s10910-015-0480-z.
G. E. Mehak and A. A. Bhatti, “Forgotten topological index of line graphs of some chemical structures in drugs,” Acta Chemica Iasi, 2018.
M. Cancan, M. Imran, S. Akhter, M. K. Siddiqui, and M. F. Hanif, “Computing forgotten topological index of extremal cactus chains,” Applied Mathematics and Nonlinear Sciences, vol. 6, no. 1, pp. 439–446, 2021. https://doi.org/10.2478/amns.2020.2.00075.
A. Gursoy, N. K. Gursoy, and A. Ulker, “Computing forgotten topological index of zero-divisor graphs of commutative rings,” Turkish Journal of Mathematics, vol. 46, no. 5, pp. 1845–1863, 2022. https://doi.org/10.55730/1300-0098.3236.