Karafan Journal

Karafan Journal

Investigating the metric dimension of an intersection graph in a commutative ring

Document Type : Original Article

Author
Reza Nikandish
Assistant Professor, Department of Mathematics, Faculty of Science, Jundi-Shapur University of Technology, Dezful, Iran.
10.48301/kssa.2021.128394
Abstract
Suppose R is a uniform commutative ring. The R-dependent intersection graph, represented by the symbol G (R), is a simple, directionless graph whose set of vertices is the set of all non-trivial ideals of R and two distinct vertices 𝐼, 𝐽 are joined if and only if 𝐼 ∩ 𝐽 ≠ (0). In this paper, the metric dimension of intersection graphs associated with commutative rings is examined and some metric dimension formulas for intersection graphs are provided.
Keywords
Metric dimension
Resolving set
Metric basis
Intersection graph
Ideal
Commutative ring
Subjects
References
[1] Nikandish, R., Nikmehr, M. J., & Bakhtyiari, M. (2015). Coloring of The Annihilator Graph of a Commutative ring. Journal of Algebra and Its Applications, 15(7), 1650124. https://doi.org/10.1142/S0219498816501243
[2] Nikmehr, M., Nikandish, R., & Bakhtyiari, M. (2017). More on the annihilator graph of a commutative ring. Hokkaido Mathematical Journal, 46(1), 107-118. https://doi. org/10.1142/S0219498819501603
[3] Pirzada, S., & Raja, R. (2017). On the metric dimension of a zero-divisor graph. Communications in Algebra, 45(4), 1399-1408. https://doi.org/10.1080/00927872. 2016.1175602
[4] Pirzada, S., Raja, R., & Redmond, S. (2014). Locating sets and numbers of graphs associated to commutative rings. Journal of Algebra and Its Applications, 13(07), 1450047. https://doi.org/10.1142/S0219498814500479
[5] Harary, F., & Melter, R. A. (1976). On the Metric Dimension of a Graph. Ars Combinatoria, 2, 191-195. https://www.scirp.org/(S(351jmbntvnsjt1aadkposzje))/ reference/ReferencesPapers.aspx?ReferenceID=388665
[6] Sharp, R. Y. (2000). Steps in Commutative Algebra (2nd ed.). London Mathematical Society student texts, Cambridge University Press. https://esploro.libs.uga.edu/disc overy/fulldisplay?vid=01GALI_UGA:UGA&docid=alma9926528343902959&context=L
[7] West, D. B. (2001). Introduction to Graph Theory. Prentice Hall. https://books.google. com/books?id=TuvuAAAAMAAJ
[8] Chakrabarty, I., Ghosh, S., Mukherjee, T. K., & Sen, M. K. (2009). Intersection graphs of ideals of rings. Discrete Mathematics, 309(17), 5381-5392. https://doi.org/10.10 16/j.disc.2008.11.034
Karafan Journal
Volume 17, Issue 4 - Serial Number 50
Technical and Engineering
Winter 2021
Pages 35-44

  • Receive Date 28 January 2020
  • Revise Date 21 March 2021
  • Accept Date 18 January 2021