On the Forgotten Index and Jacobson Graphs Associated with Integer Rings Modulo n


Mohammad Nafie Jauhari(1*), Shahnaz Latifatul Jannah(2), Turmudi Turmudi(3), Intan Nisfulaila(4)

(1) Universitas Islam Negeri Maulana Malik Ibrahim Malang, Indonesia, Indonesia
(2) Universitas Islam Negeri Maulana Malik Ibrahim Malang, Indonesia, Indonesia
(3) Universitas Islam Negeri Maulana Malik Ibrahim Malang, Indonesia, Indonesia
(4) Universitas Islam Negeri Maulana Malik Ibrahim Malang, Indonesia, Indonesia
(*) Corresponding Author

Abstract


This paper investigates the connections between the Jacobson graph and the algebraic properties of rings through the analysis of the Jacobson graph of the ring \mathbb{Z}_{3p}, where p is a prime number greater than 3. The Jacobson graph of a commutative ring R is constructed by taking the elements of R, excluding its Jacobson Radical, as vertices, and connecting two distinct vertices if 1 minus their product is not a unit in R. The F-Index is utilized to capture and represent the structural properties of the ring through its associated graph. A detailed examination of the Jacobson Radical, maximal ideals, and vertex degrees in \mathbb{Z}_{3p} leads to the calculation of the F-Index, providing insights into the graph’s connectivity and underlying algebraic structure. This study contributes to the intersection of algebra and graph theory, offering a foundation for further research into more complex algebraic structures.

Keywords


F-Index, Jacobson Graph, Jacobson Radical, Maximal ideal, Modulo Ring

Full Text:

PDF

References


S. Pirzada, “Applications of Graph Theory,” Pamm, vol. 7, no. 1, p. 2070013, 2007.

F. Afzal, F. Asmat, and D. Afzal, “Molecular Description for Magnesium Iodide,” Math. Model. Eng., vol. 5, no. 4, pp. 175–189, 2019.

S. Wang and B. Wei, “Multiplicative Zagreb Indices of Cacti,” Discret. Math. Algorithms Appl., vol. 08, no. 03, p. 1650040, 2016.

A. Azimi and A. Erfanian, “The Jacobson Graph of Commutative Rings,” J. Algebr. Its Appl., vol. 12, no. 03, p. 1250179, 2012.

S. Humaira, P. Astuti, I. Muchtadi-Alamsyah, and A. Erfanian, “The Matrix Jacobson Graph of Fields,” J. Phys. Conf. Ser., vol. 1538, no. 1, p. 12008, 2020.

A. Azimi and A. Erfanian, “Isomorphisms Between Jacobson Graphs,” Rend. Del Circ. Mat. Di Palermo (1952 -), vol. 63, no. 2, pp. 277–286, 2014.

B. Furtula and I. Gutman, “A Forgotten Topological Index,” J. Math. Chem., vol. 53, no. 4, pp. 1184–1190, 2015.

A. Jahanbani, M. Atapour, and R. Khoeilar, “New Results on the Forgotten Topological Index and Coindex,” J. Math., vol. 2021, pp. 1–11, 2021.

S. Madhumitha, “Graphs Defined on Rings: A Review,” Mathematics, vol. 11, no. 17, p. 3643, 2023.

S. Elumalai, T. Mansour, and M. A. Rostami, “On the Bounds of the Forgotten Topological Index,” Turkish J. Math., vol. 41, pp. 1687–1702, 2017.

N. Akgüneş, “Analyzing Special Parameters Over Zero-Divisor Graphs,” 2012.

S. Aykaç and N. Akgüneş, “Analysis of Graph Parameters Associated With Zero-Divisor Graphs of Commutative Rings,” New Trends Math. Sci., vol. 2, no. 6, pp. 144–149, 2018.




DOI: https://doi.org/10.15575/kubik.v9i2.39585

Refbacks

  • There are currently no refbacks.


Copyright (c) 2024 Mohammad Nafie Jauhari

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.


Journal KUBIK: Jurnal Publikasi Ilmiah Matematika has indexed by:

SINTA DOAJ Dimensions Google Scholar Garuda Moraref DOI Crossref

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.