Metric Dimension of Zero-Divisor Graph for the Ring Zn

Mithun Basak¹ , Laxman Saha² , Kalishankar Tiwary³

Section:Research Paper, Product Type: Journal-Paper
Vol.6 , Issue.6 , pp.74-78, Dec-2019

Online published on Dec 31, 2019

Copyright © Mithun Basak, Laxman Saha, Kalishankar Tiwary . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
Metric Dimension of a simple connected graph is the minimum number of vertices those are used to identify each vertex of the graph uniquely using distance code. In this paper, we determine metric dimension of zero-divisor graph associated with the ring Zn.

Key-Words / Index Term :
Ring, Metric Dimension, Zero divisors, Zero-divisor Graph

References :
[1]Akbari, S., Mohammadian, A. On the zero-divisor graph of finite rings. J. Algebra. 314(1), 168–184, 2007
[2] Anderson, D. F., Frazier, A., Lauve, A., Livingston, P. S. The Zero-Divisor Graph of a
Commutative Ring, II, Lecture Notes in Pure and Applied Mathematics. 220, 61–72, 2001
[3] Anderson, D. F., Levy, R., Shapiro, J. Zero-divisor graphs, von Neumann regular rings, and Boolean algebras. J. Pure Appl. Algebra. 180, 221–241, 2003
[4] Anderson, D. F., Livingston, P. S. The zero-divisor graph of a commutative ring. J.Algebra. 217 , 434–447, 1999
[5] Anderson, D. D., Naseer, M. (1993). Beck‘s coloring of a commutative ring. J. Algebra. 159, 500–517, 1993
[6] Atiyah, M. F., MacDonald, I. G. Introduction to Commutative Algebra. Reading, MA: Addison-Wesley, 1969
[7] Beck, I. Coloring of commutative rings. J. Algebra. 116, 208–226, 1988
[8] Beerliova, Z., Eberhard, F., Erlebach, T., Hall, A., Ho.mann, M., Mihalak, M., Ram, L. S. Network discovery and verification. IEEE J. On Sel. Areas Commun. 24, 2168–2181, 2006
[9] Chvatal, V. Mastermind. Combinatorica. 3, 325–329, 1983
[10] Corbas, B., Williams, G. D. Rings of order p5 II. Local rings. J. Algebra. 231(2), 691–704, 2000
[11] Garey, M. R., Johnson, D. S. Computers and Intractability. A Guide to the Theory of NP-Completeness. New York: Freeman 1979.
[12] Harary, F., Melter, R. A. On the metric dimension of a graph. Ars Combin. 2, 191–195, 1976
[13] Khuller, S., Raghavachari, B. and Rosenfeld, A. Landmarks in graphs, Discrete Appl.Math., 70, 217–229, 1996
[14] Mulay, S. B. Cycles and symmetries of zero-divisors. Commun. Algebra. 30(7), 3533–3558.
[15] Pirzada, S., Raja, R., Redmond, S. P. Locating sets and numbers of graphs associated to commutative rings. J. Algebra Appl. 13(7), 1450047 18 pp, 2014
[16] Raja, R., Pirzada, S., Redmond, S. P. On Locating numbers and codes of zero-divisor graphs associated with commutative rings. J. Algebra Appl. 15(1), 1650014 22pp, 2016
[17] Redmond, S. P. The zero-divisor graph of a noncommutative ring. Int. J. Commutative Rings. 1(4) , 203–211, 2002
[18] Redmond, S. P. An ideal-based zero-divisor graph of a commutative ring. Commun. Algebra. 31, 4425–4443, 2003
[19] Sebo, A., Tannier, E. On metric generators of graphs. Math. Oper. Res. 29, 383–393, 2004
[20] Slater, P. Leaves of trees. Proc. 6th Southeastern Conf. on Combinatorics, Graph Theory and Computing, 14, 549–559, 1975