Computing metric dimension of compressed zero divisor graphs associated to rings

[1] S. Akbari and A. Mohammadian, On zero divisor graphs of a ring, J. Algebra, 274 (2004), 847–855.10.1016/S0021-8693(03)00435-6Search in Google Scholar

[2] D. D. Anderson and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra, 159 (1993), 500–514.10.1006/jabr.1993.1171Search in Google Scholar

[3] D. F. Anderson and P. S. Livingston, The zero divisor graph of a commutative ring, J. Algebra, 217 (1999), 434–447.10.1006/jabr.1998.7840Search in Google Scholar

[4] D. F. Anderson and J. D. LaGrange, Commutative Boolean Monoids, reduced rings and the compressed zero-divisor graphs, J. Pure and Appl. Algebra, 216 (2012), 1626–1636.10.1016/j.jpaa.2011.12.002Search in Google Scholar

[5] D. F. Anderson and J.D. LaGrange, Some remarks on the compressed zero-divisor graphs, J. Algebra, 447 (2016), 297–321.10.1016/j.jalgebra.2015.08.021Search in Google Scholar

[6] M. F. Atiyah and I. G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, MA (1969).Search in Google Scholar

[7] I. Beck, Coloring of commutative rings, J. Algebra, 26 (1988), 208–226.10.1016/0021-8693(88)90202-5Search in Google Scholar

[8] P. J. Cameron and J. H. Van Lint, Designs, Graphs, Codes and their Links, in: London Mathematical Society Student Texts, 24 (5), Cambridge University Press, Cambridge, 1991.10.1112/blms/24.5.500aSearch in Google Scholar

[9] G. Chartrand, L. Eroh, M. A. Johnson and O. R. Oellermann, Resolvability in graphs and metric dimension of a graph, Disc. Appl. Math., 105 (1-3) (2000), 99–113.10.1016/S0166-218X(00)00198-0Search in Google Scholar

[10] B. Corbas and G. D. Williams, Rings of order p⁵. II. Local rings, J. Algebra, 231 (2) (2000), 691–704.10.1006/jabr.2000.8350Search in Google Scholar

[11] R. Diestel, Graph Theory, 4th ed. Vol. 173 of Graduate texts in mathematics, Heidelberg: Springer, 2010.10.1007/978-3-642-14279-6Search in Google Scholar

[12] F. Harary and R. A. Melter, On the metric dimension of a graph, Ars Combin., 2 (1976), 191–195.Search in Google Scholar

[13] J. A. Huckaba, Commutative Rings with Zero Divisors, Marcel-Dekker, New York, Basel, 1988.Search in Google Scholar

[14] S. Khuller, B. Raghavachari and A. Rosenfeld, Localization in graphs, Technical Report CS-TR-3326, University of Maryland at College Park, 1994.Search in Google Scholar

[15] H. R. Maimani, M. R. Pournaki and S. Yassemi, Zero divisor graph with respect to an ideal, Comm. Algebra, 34 (3) (2006), 923–929.10.1080/00927870500441858Search in Google Scholar

[16] S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra, 30 (7) (2002), 3533–3558.10.1081/AGB-120004502Search in Google Scholar

[17] S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient Blackswan, Hyderabad, 2012.Search in Google Scholar

[18] S. Pirzada, Rameez Raja and S. P. Redmond, Locating sets and numbers of graphs associated to commutative rings, J. Algebra Appl., 13 (7) (2014), 1450047.10.1142/S0219498814500479Search in Google Scholar

[19] S. Pirzada and Rameez Raja, On the metric dimension of a zero divisor graph, Comm. Algebra, 45 (4) (2017), 1399–1408.10.1080/00927872.2016.1175602Search in Google Scholar

[20] S. Pirzada and Rameez Raja, On graphs associated with modules over commutative rings, J. Korean Math. Soc., 53 (5) (2016), 1167–1182.10.4134/JKMS.j150457Search in Google Scholar

[21] Rameez Raja, S. Pirzada and Shane Redmond, On locating numbers and codes of zero divisor graphs associated with commutative rings, J. Algebra Appl., 15 (1) (2016), 1650014.10.1142/S0219498816500146Search in Google Scholar

[22] S. P. Redmond, On zero divisor graph of small finite commutative rings, Disc. Math., 307 (2007), 1155–1166.10.1016/j.disc.2006.07.025Search in Google Scholar

[23] S. P. Redmond, An Ideal based zero divisor graph of a commutative ring, Comm. Algebra, 31 (9) (2003), 4425–4443.10.1081/AGB-120022801Search in Google Scholar

[24] A. Sebo and E. Tannier, On metric generators of graphs, Math. Oper. Research., 29 (2) (2004), 383–393.10.1287/moor.1030.0070Search in Google Scholar

[25] P. J. Slater, Dominating and reference sets in a graph, J. Math, Phys. Sci., 22 (1988), 445–455.Search in Google Scholar

[26] P.J. Slater, Leaves of trees, Congressus Numerantium, 14 (1975), 549–559.Search in Google Scholar

[27] S. Spiroff and C. Wickham, A zero divisor graph determined by equivalence classes of zero divisors, Comm. Algebra, 39 (7) (2011), 2338–2348. 10.1080/00927872.2010.488675Search in Google Scholar