The Metric Dimension and Local Metric Dimension of Relative Prime Graph

Inna Kuswandari, Fatmawati Fatmawati, Mohammad Imam Utoyo

Abstract


This study aims to determine the value of metric dimensions and local metric dimensions of relative prime graphs formed from modulo  integer rings, namely . As a vertex set is  and  if  and  are relatively prime. By finding the pattern elements of resolving set and local resolving set, it can be shown the value of the metric dimension and the local metric dimension of graphs  are  and  respectively, where  is the number of vertices groups that formed multiple 2,3, … ,  and  is the cardinality of set . This research can be developed by determining the value of the fractional metric dimension, local fractional metric dimension and studying the advanced properties of graphs related to their forming rings.

Key Words : metric dimension; modulo ; relative prime graph; resolving set; rings.

Keywords


Graph Theory

Full Text:

PDF

References


G. Chartrand and L. Lesniak, Graphs and Digraphs, Third Edition, Chapman & Hall/CRC, Florida, 2000.

P. J. Slater, "Leaves and trees", Congr. Numer., 14, pp.549-559, 1975.

F. Harary and R. A. Melter, "On the Metric Dimension of a Graph", Ars Combinatoria, Vol. 2, pp. 191-195, 1976.

J. A. Rodriquez-Velazquez, I. G. Yero, D. Kuziak and O. R. Oellermann, "On the strong metric dimension of cartesian and direct product of graphs", Discrete Mathematics 335, pp. 8-19, 2014.

F. Okamoto, B. Phinezy and P. Zhang, "The Local Metric Dimension of a graph", Math. Bohem., Vol. 135, pp. 239-255, 2010.

I. Beck, "Coloring of Commutative Rings", Journal of Algebra, Vol. 116, No.1, pp. 208-226, 1988.

D. F. Anderson and P. S. Livingston, "The Zero-Divisor Graph of a Commutative Ring", Journal of Algebra 217, pp. 434-447 1999.

S. P. Redmond, "The zero-divisor graph of a non-commutative ring", International J. Commutative Rings 1(4), pp. 203-211, 2002.

S. P. Redmond, "Central Sets and Radii of the Zero-Divisor Graphs of Commutative Rings", Communications in Algebra 34, pp. 2389–2401, 2006.

A. Azimi, A. Erfanian and D. G. Farrokhi, "The Jacobson Graph of Commutative Rings", Jounal of Algebra and its Application, DOI: 10.1142/S0219498812501794, 2012.

A. Novictor, L. Susilowati and Fatmawati, "Jacobson graph construction of ring Z_(3^n ), for n>1", Journal of Physics: Conference Series 1494 (2020) 012016, doi:10.1088/1742-6596/1494/1/012016, 2020.

J. B. Fraleigh, A First Course in Abstract Algebra, Addison-Wesley Publishing Company, Massachusetts, 2003.

G. Chartrand, L. Eroh, M. A. Johnson and O. R. Oellermann, "Resolvability in graphs and the metric dimension of a graph", Discrete Applied Mathematics 105, pp. 99-113, 2000.




DOI: https://doi.org/10.18860/ca.v6i3.10103

Refbacks

  • There are currently no refbacks.


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

Editorial Office
Mathematics Department,
Universitas Islam Negeri Maulana Malik Ibrahim Malang
Jalan Gajayana 50 Malang, Jawa Timur, Indonesia 65144
Faximile (+62) 341 558933
e-mail: cauchy@uin-malang.ac.id

Creative Commons License
CAUCHY: Jurnal Matematika Murni dan Aplikasi is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.