Hostname: page-component-7c8c6479df-995ml Total loading time: 0 Render date: 2024-03-28T11:33:26.076Z Has data issue: false hasContentIssue false

THE METRIC DIMENSION OF THE ANNIHILATING-IDEAL GRAPH OF A FINITE COMMUTATIVE RING

Published online by Cambridge University Press:  27 April 2021

DAVID DOLŽAN*
Affiliation:
Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 21, SI-1000Ljubljana, Slovenia

Abstract

We determine the metric dimension of the annihilating-ideal graph of a local finite commutative principal ring and a finite commutative principal ring with two maximal ideals. We also find bounds for the metric dimension of the annihilating-ideal graph of an arbitrary finite commutative principal ring.

Type
Research Article
Copyright
© 2020 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The author acknowledges the financial support from the Slovenian Research Agency (research core funding no. P1-0222).

References

Aalipour, G., Akbari, S., Behboodi, M., Nikandish, R., Nikmehr, M. J. and Shaveisi, F., ‘The classification of the annihilating-ideal graphs of commutative rings’, Algebra Colloq. 21(2) (2014), 249256.10.1142/S1005386714000200CrossRefGoogle Scholar
Aijaz, M. and Pirzada, S., ‘Annihilating-ideal graphs of commutative rings’, Asian-Eur. J. Math. 13(7) (2020), Article ID 2050121.CrossRefGoogle Scholar
Ali, F., Salman, M. and Huang, S., ‘On the commuting graph of dihedral group’, Comm. Algebra 44(6) (2016), 23892401.10.1080/00927872.2015.1053488CrossRefGoogle Scholar
Aliniaeifard, F., Behboodi, M. and Li, Y., ‘The annihilating-ideal graph of a ring’, J. Korean Math. Soc. 52(6) (2015), 13231336.CrossRefGoogle Scholar
Anderson, D. F. and Badawi, A., ‘The total graph of a commutative ring’, J. Algebra 320 (2008), 27062719.10.1016/j.jalgebra.2008.06.028CrossRefGoogle Scholar
Anderson, D. F. and Livingston, P. S., ‘The zero-divisor graph of a commutative ring’, J. Algebra 217 (1999), 434447.CrossRefGoogle Scholar
Beck, I., ‘Coloring of commutative rings’, J. Algebra 116 (1988) 208226.CrossRefGoogle Scholar
Behboodi, M. and Rakeei, Z., ‘The annihilating-ideal graph of commutative rings I’, J. Algebra Appl. 10(4) (2011), 727739.10.1142/S0219498811004896CrossRefGoogle Scholar
Behboodi, M. and Rakeei, Z., ‘The annihilating-ideal graph of commutative rings II’, J. Algebra Appl. 10(4) (2011), 741753.CrossRefGoogle Scholar
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).CrossRefGoogle Scholar
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), 99113.CrossRefGoogle Scholar
Curtis, A. R., Diesl, A. J. and Rieck, J. C., ‘Classifying annihilating-ideal graphs of commutative Artinian rings’, Comm. Algebra 46(9) (2018), 41314147.CrossRefGoogle Scholar
Díaz, J., Pottonen, O., Maria, M. and van Leeuwen, E. J., ‘On the complexity of metric dimension’, in: Algorithms–ESA 2012, Lecture Notes in Computer Science, 7501 (Springer, Heidelberg, 2012), 419430.CrossRefGoogle Scholar
Dolžan, D., ‘The metric dimension of the total graph of a finite commutative ring’, Canad. Math. Bull. 59(4) (2016), 748759.CrossRefGoogle Scholar
Epstein, L., Levin, A. and Woeginger, G. J., ‘The (weighted) metric dimension of graphs: hard and easy cases’ (English summary), in: Graph-Theoretic Concepts in Computer Science, Lecture Notes in Computer Science, 7551 (Springer, Heidelberg, 2012), 114125.10.1007/978-3-642-34611-8_14CrossRefGoogle Scholar
Harary, F. and Melter, R. A., ‘On the metric dimension of a graph’, Ars. Combin. 2 (1976), 191195.Google Scholar
Khuller, S., Raghavachari, B. and Rosenfeld, A., ‘Localization in graphs’, Technical Report CS-TR-3326, Department of Computer Science, University of Maryland at College Park, 1994.Google Scholar
McDonald, B. R., Finite Rings with Identity, Pure and Applied Mathematics, 28 (Marcel Dekker, New York, 1974).Google Scholar
Nikandish, R., Maimani, H. R. and Kiani, S., ‘Domination number in the annihilating-ideal graphs of commutative rings’, Publ. Inst. Math. (Beograd) (N.S.) 97(111) (2015), 225231.CrossRefGoogle Scholar
Pirzada, S., Aijaz, M. and Redmond, S., ‘Upper dimension and bases of zero divisor graphs of commutative rings’, AKCE Int. J. Graphs Comb. 17(1) (2020), 168173.10.1016/j.akcej.2018.12.001CrossRefGoogle Scholar
Pirzada, S. and Raja, R., ‘On the metric dimension of a zero-divisor graph’, Comm. Algebra 45(4) (2017), 13991408.CrossRefGoogle Scholar
Sebö, A. and Tannier, E., ‘On metric generators of graphs’, Math. Oper. Res. 29 (2004), 383393.10.1287/moor.1030.0070CrossRefGoogle Scholar
Shaveisi, F., ‘Some results on the annihilating-ideal graphs’, Canad. Math. Bull. 59(3) (2016), 641651.CrossRefGoogle Scholar
Slater, P. J., ‘Leaves of trees’, Congr. Numer. 14 (1975), 549559.Google Scholar
Tamizh Chelvam, T. and Selvakumar, K., ‘On the connectivity of the annihilating-ideal graphs’, Discuss. Math. Gen. Algebra Appl. 35(2) (2015), 195204.Google Scholar