The First Zagreb Index of the Zero Divisor Graph for the Ring of Integers Modulo Power of Primes

Authors

  • Ghazali Semil @ Ismail ᵃDepartment of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia; ᵇMathematical Sciences Studies, College of Computing, Informatics and Mathematics, Universiti Teknologi MARA Johor Branch, Pasir Gudang Campus, 81750, Masai, Johor, Malaysia
  • Nor Haniza Sarmin Department of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia
  • Nur Idayu Alimon Mathematical Sciences Studies, College of Computing, Informatics and Mathematics, Universiti Teknologi MARA Johor Branch, Pasir Gudang Campus, 81750, Masai, Johor, Malaysia
  • Fariz Maulana Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, 40132, Bandung, Indonesia

DOI:

https://doi.org/10.11113/mjfas.v19n5.2980

Keywords:

Topological index, first Zagreb index, zero divisor graph, graph theory, ring theory

Abstract

Let  be a simple graph with the set of vertices and edges. The first Zagreb index of a graph is defined as the sum of the degree of each vertex to the power of two. Meanwhile, the zero divisor graph of a ring , denoted by , is defined as a graph with its vertex set  contains the nonzero zero divisors in which two distinct vertices  and  are adjacent if . In this paper, the general formula of the first Zagreb index of the zero divisor graph for the commutative ring of integers modulo ,  where a prime number  and a positive integer  is determined. A few examples are given to illustrate the main results.                                                                           

Author Biographies

Ghazali Semil @ Ismail, ᵃDepartment of Mathematical Sciences, Faculty of Science, Universiti Teknologi Malaysia, 81310 UTM Johor Bahru, Johor, Malaysia; ᵇMathematical Sciences Studies, College of Computing, Informatics and Mathematics, Universiti Teknologi MARA Johor Branch, Pasir Gudang Campus, 81750, Masai, Johor, Malaysia

 

 

 

Nur Idayu Alimon, Mathematical Sciences Studies, College of Computing, Informatics and Mathematics, Universiti Teknologi MARA Johor Branch, Pasir Gudang Campus, 81750, Masai, Johor, Malaysia

 

 

Fariz Maulana, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, 40132, Bandung, Indonesia

 

 

References

Trinajstic, N. (2018). Chemical Graph Theory. Routledge. https://doi.org/10.1201/9781315139111.

Basak, S. C., Magnuson, V. R., Niemi, G. J., Regal, R. R., & Veith, G. D. (1987). Topological indices: Their nature, mutual relatedness, and applications. Mathematical Modelling, 8, 300-305. https://doi.org/10.1016/0270-0255(87)90594-X.

Darafsheh, M. R. (2010). Computation of topological indices of some graphs. Acta Applicandae Mathematicae, 110, 1225-1235.

Gutman, I., & Das, K. C. (2004). The first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem, 50(1), 83-92.

Ghorbani, M., & Hosseinzadeh, M. (2012). A new version of Zagreb indices. Filomat, 26(1), 93-100. https://doi.org/10.2298/FIL1201093G.

Alimon, N. I., Sarmin, N. H., & Erfanian, A. (2018). Topological indices of non-commuting graph of dihedral groups. Malaysian Journal of Fundamental and Applied Sciences, 473-476.

Das, K. Ch., Xu, K., & Nam, J. (2015). Zagreb indices of graphs. Frontiers of Mathematics in China, 10(3), 567-582. https://doi.org/10.1007/s11464-015-0431-9.

Aykaç, S., Akgüneş, N., & Çevik, A. S. (2019). Analysis of Zagreb indices over zero-divisor graphs of commutative rings. Asian-European Journal of Mathematics, 12(06), 2040003. https://doi.org/10.1142/S1793557120400033.

Mazlan, N. ‘A., Mat Hassim, H. I., Sarmin, N. H., & Salih Khasraw, S. M. (2022). The first Zagreb index of zero-divisor type graph for some rings of integers modulo n. Proc. Sci. Math., 29-32.

Pattabiraman, K., & Suganya, T. (2021). Zagreb indices of vertex based on IB product. Journal of Physics: Conference Series, 1724(1), 012037. https://doi.org/10.1088/1742-6596/1724/1/012037.

Rayer, C. J., & Jeyaraj, R. S. (2023). Applications on topological indices of zero-divisor graph associated with commutative rings. Symmetry, 15(2), 335. https://doi.org/10.3390/sym15020335.

Anderson, D. F., & Livingston, P. S. (1999). The zero-divisor graph of a commutative ring. Journal of Algebra, 217(2), 434-447. https://doi.org/10.1006/jabr.1998.7840.

Shuker, N. H., & Rashed, P. A. (2015). The zero divisor ideal graph of the ring Z_n. International Journal of Advance Research (IJOAR). 3(5), 1-12.

Smith, B. (2016). Perfect zero-divisor graphs of Z_n. Rose-Hulman Undergraduate Mathematics Journal, 17(2), 6.

Seeta, V. (2021). Zero divisor graph of a commutative ring. Turkish Online Journal of Qualitative Inquiry, 12(7).

Anderson, D. F., & Weber, D. (2018). The zero-divisor graph of a commutative ring without identity. International Electronic Journal of Algebra, 176-202. https://doi.org/10.24330/ieja.373663.

Singh, P., & Bhat, V. K. (2020). Zero-divisor graphs of finite commutative rings: A survey. Surveys in Mathematics & Its Applications, 15.

Coykendall, J., Sather-Wagstaff, S., Sheppardson, L., & Spiroff, S. (2012). On zero divisor graphs. Progress in Commutative Algebra, 2, 241-299.

Baruah, D. (2017). Construction of zero Divisor graphs of rings. International Journal of Mathematics Trends and Technology, 48(3), 180-185. https://doi.org/10.14445/22315373/IJMTT-V48P525.

Zaid, N., Sarmin, N. H., & Khasraw, S. M. S. (2020). The applications of zero divisors of some finite rings of matrices in probability and graph theory. Jurnal Teknologi, 83(1), 127-132. https://doi.org/10.11113/jurnalteknologi.v83.14936.

Kuppan, A., & Ravi Sankar, J. (2022). Prime decomposition of zero divisor graph in a commutative ring. Mathematical Problems in Engineering, 2022, 1-4. https://doi.org/10.1155/2022/2152513.

Wisbauer, R. (2018). Foundations of module and ring theory. Routledge.

Rowen, L. H. (1988). Ring theory V1. Academic Press.

Fraleigh, J. B. (2003). A first course in abstract algebra. Pearson Education India.

Wilson, R. J. (2015). Introduction to graph theory. PDF eBook. Pearson Higher Ed.

Gunderson, D. S. (2014). Handbook of Mathematical Induction. Chapman and Hall/CRC. https://doi.org/10.1201/b16005.

Juliana, R. (2022). Karakteristik graf pembagi nol pada gelanggang bilangan bulat modulo. Fraktal: Jurnal Matematika dan Pendidikan Matematika, 3(2), 1-8.

Downloads

Published

19-10-2023