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The Metric Dimension of the Total Graph of a Finite Commutative Ring

Published online by Cambridge University Press:  20 November 2018

David Dolžan
David Dolžan*
Affiliation:
Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia e-mail: [email protected]
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Abstract

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We study the total graph of a finite commutative ring. We calculate its metric dimension in the case when the Jacobson radical of the ring is nontrivial, and we examine the metric dimension of the total graph of a product of at most two fields, obtaining either exact values in some cases or bounds in other, depending on the number of elements in the respective fields.

Keywords

13M9905E40total graphfinite ringmetric dimension
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 4 , 01 December 2016 , pp. 748 - 759
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Akbari, S., Kiani, D., Mahammadi, F., and Moradi, S., The total graph and regular graph of a commutative ring. J. Pure Appl. Algebra 213(2009), no. 12, 2224–2228. http://dx.doi.Org/10.1016/j.jpaa.2009.03.013 Google Scholar
[2] Diaz, J., Pottonen, O., Maria, M., and van Leeuwen, E. J., On the complexity of metric dimension. In: Algorithms-ESA 2012, Lecture Notes in Comput. Sci., 7501, Springer, Heidelberg, 2012, pp. 419–430. http://dx.doi.Org/10.1007/978-3-642-33090-2_37 Google Scholar
[3] Epstein, L., Levin, A., and Woeginger, G. J., The (weighted) metric dimension of graphs: hard and easy cases. In: Graph-theoretic concepts in computer science, Lecture Notes in Comput. Sci., 7551, Springer, Heidelberg, 2012, pp. 114–125. http://dx.doi.Org/10.1007/978-3-642-34611-8_14 Google Scholar
[4] Ali, F., Salman, M., and Huang, S., On the commuting graph of dihedral group. Comm. Algebra, to appear.Google Scholar
[5] Anderson, D. F. and Badawi, A., The total graph of a commutative ring. J. Algebra 320(2008), no. 7, 2706–2719. http://dx.doi.Org/10.1016/j.jalgebra.2008.06.028 Google Scholar
[6] Cameron, P. J. and Van Lint, J. H., Designs, graphs, codes and their links. London Mathematical Society Student Texts, 22, Cambridge University Press, Cambridge, 1991. http://dx.doi.Org/10.1017/CBO9780511623714 Google Scholar
[7] Chartrand, G., Eroh, L., Johnson, M. A., and Oellermann, O. R., Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105(2000), no. 1-3, 99–113. http://dx.doi.Org/10.1016/S0166-218X(00)00198-0 Google Scholar
[8] Dolžan, D. and Oblak, P., The total graph of finite rings. Comm. Algebra 43(2015), no. 7, 2903–2911. http://dx.doi.Org/10.1080/00927872.2014.90741 7 Google Scholar
[9] Ganesan, N., Properties of rings with a finite number of zero divisors. Math. Ann. 161(1965), 241–246. http://dx.doi.Org/10.1007/BF01359907 Google Scholar
[10] Harary, F. and Melter, R. A., On the metric dimension of a graph. Ars. Combinatoria 2(1976), 191–195.Google Scholar
[11] Hernando, C., Mora, M., Pelayo, I., Seara, C., and Wood, D. R., Extremal graph theory for metric dimension and diameter. Electron. J. Combin. 17(2010), no. 1, Research Paper 30.Google Scholar
[12] Khuller, S., Raghavachari, B., and Rosenfeld, A., Localization in graphs. Technical report CS-TR-3326, University of Maryland at College Park, 1994.Google Scholar
[13] Koh, K., On properties of rings with a finite number of zero divisors. Math. Ann. 171(1967), 79–80. http://dx.doi.Org/10.1007/BF01433095 Google Scholar
[14] Maimani, H. R., Wickham, C., and Yassemi, S., Rings whose total graphs have genus at most one. Rocky Mountain J. Math 42(2012), no. 5,1551-1560. http://dx.doi.Org/10.1216/RMJ-2O12-42-5-1551 Google Scholar
[15] Sebo, A. and Tannier, E., On metric generators of graphs. Math. Oper. Res. 29(2004), no. 2, 383–393. http://dx.doi.Org/10.1287/moor.1030.0070 Google Scholar
[16] Shekarriz, M. H., Shirdareh Haghighi, M. H., and Sharif, H., On the total graph of a finite commutative ring. Comm. Algebra 40(2012), no. 8, 2798–2807. http://dx.doi.Org/10.1080/00927872.2011.585680 Google Scholar
[17] Slater, P. J., Leaves of trees. In: Proceedings of the Sixth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1975), Congressus Numerantium, 14, Utilitas Math., Winnipeg, MB, 1975, pp. 549–559.Google Scholar
[18] Tamizh, T. Chelvam and Asir, T., On the genus of the total graph of a commutative ring. Comm. Algebra 41(2013), no. 1, 142–153. http://dx.doi.Org/10.1080/00927872.2011.624147 Google Scholar