.svg){kind=logo}
ON METRIC DIMENSION AND RESOLVING-DOMINATION NUMBER OF ZERO-DIVISOR GRAPHS OF DIRECT PRODUCT OF FINITE FIELDS
Keywords:
zero-divisor graphs, metric dimension, resolving domination number.DOI:
https://doi.org/10.17654/0972555523016Abstract
The main objective of this paper is to calculate the metric dimension and resolving-domination number of zero-divisor graphs of direct product of finite fields. Let $F_1, \ldots, F_n(n \geqslant 2)$ be finite fields. We consider the zero-divisor graph $\Gamma(R)$ of the ring $R=F_1 \times \cdots \times F_n$ $(n \geqslant 2)$ of direct product of finite fields. We determine metric dimension $\beta(\Gamma(R))$ and resolving-domination number $\gamma_r(\Gamma(R))$ of the ring of direct product of finite fields. If the order of each field is two, then we prove that the metric dimension $\beta(\Gamma(R)) \leqslant n-1$ and resolving-domination number $\gamma_r(\Gamma(R))=n$. If the order of each field is at least 3 , then we prove that the metric dimension $\beta(\Gamma(R))$ and resolving-domination number $\gamma_r(\Gamma(R))$ are both equal to $|V(\Gamma(R))|-\left(2^n-2\right)$.
Received: February 19, 2023
Revised: July 1, 2023
Accepted: July 4, 2023
References
D. F. Anderson, T. Asir, A. Badawi and T. T. Chelvam, Graphs from Rings, Springer, 2021.
D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217(2) (1999), 434-447.
I. Beck, Coloring of commutative rings, J. Algebra 116(1) (1988), 208-226.
G. Chartrand, L. Eroh, M. A. Johnson and O. R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105(1-3) (2000), 99-113.
J. Currie and O. R. Oellerman, The metric dimension and metric independence of a graph, J. Combin. Math. Combin. Comput. 39 (2001), 157-167.
F. Harary and R. A. Melter, On the metric dimension of a graph, Ars Combin. 2 (1976), 191-195.
N. J. Rad, S. H. Jafari and D. A. Mojdeh, On domination in zero-divisor graphs, Canad. Math. Bull. 56(2) (2013), 407-411.
P. J. Slater, Leaves of trees, Congr. Numer. 14 (1975), 549-559.
G. Subhash and S. N. Nithya, On connectivity of zero-divisor graphs and complement graphs of some semi-local rings, J. Comput. Math. 6(2) (2022), 135-141.
D. B. West, Introduction to Graph Theory, Vol. 2, Prentice Hall, Upper Saddle River, 2001.
Downloads
Published
Issue
Section
License
Copyright (c) 2023 PUSHPA PUBLISHING HOUSE, PRAYAGRAJ, INDIA
This work is licensed under a Creative Commons Attribution 4.0 International License.
_________________________________
Attribution: Credit Pusha Publishing House as the original publisher, including title and author(s) if applicable.
Non-Commercial Use: For non-commercial purposes only. No commercial activities without explicit permission.
No Derivatives: Modifying or creating derivative works not allowed without written permission.
Contact Pusha Publishing House for more info or permissions.
