In this paper, the twin non-commuting graph of the group G is introduced by partitioning the non-commuting graph vertices. This classification of the vertices based on special property, precisely being twin vertices property. We choose a vertex from each class as a representing vertex for the twin non-commuting graph. Moreover the adjacency of two vertices in twin non-commuting graph depends on the connectivity of them in the main non-commuting graph. We observe that the twin non-commuting graph of an AC-group is a complete graph. Moreover, some results about the metric dimension of the non-commuting graph is obtained. For instance, the metric dimension of the non-commuting graph is 3 if and only if it is associated with \(S_3,D_8,Q_8\).
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Abdollahi, A., Akbari, S., Maimani, H.R.: Non-commuting graph of a group. J. Algebra 298, 468–492 (2006)
Abdollahi, A., Jafarian Amiri, S.M., Mohammadi Hassanabadi, A.: Groups with specific number of centralizers. Houst. J. Math. 33, 43–57 (2007)
Ashrafi, A.R.: On finite groups with a given number of centralizers. Algebra Colloq. 7(2), 139–146 (2000)
Ashrafi, A.R.: Counting the centralizers of some finite groups. Korean J. Comput. Appl. Math. 7(1), 115–124 (2000)
Bagheri Gh., B., Jannesari, M., Omoomi, B.: Relations between metric dimension and domination number of graphs. arxiv:abs/1112.2326
Belcastro, S.M., Sherman, G.J.: Counting centralizers in finite groups. Math. Mag. 5, 111–114 (1994)
Bondy, J.A., Murty, J.S.R.: Graph Theory with Applications. Elsevier, Amsterdam (1977)
Chartrand, G., Eroh, L., Johnson, M.A., Ollerman, O.R.: Resolvability in graphs and the metric dimension of a graph. Discrete Appl. Math. 105, 99–113 (2000)
Darafsheh, M.R.: Groups with the same non-commuting graph. Discrete Appl. Math. 157, 833–837 (2009)
GAP: GAP-groups, algorithm and programming. http://www.gap-system.org. Accessed 7 Oct 2018
Godsil, C.: Algebric Graph Theory. Springer, New York (2001)
Hekster, N.S.: On the structure of n-isoclinism classes of groups. J. Pure Appl. Algebra 40, 63–65 (1986)
Harary, F., Melter, R.A.: On metric dimension of a graph. Ars Comb. 2, 191–195 (1976)
Hernando, C., Mora, M., Pelayo, I.M., Seara, C., Wood, D.R.: Extemal graph theory for metric dimension and diameter. Electron. J. Comb. 17, 1–28 (2010)
Itô, N.: On finite groups with given conjugate types. II. Osaka J. Math. 7, 231–251 (1970)
Moghaddamfar, A.R., Shi, W.J., Zhou, W., Zokayi, A.R.: On noncommuting graph associated with a finite group. Sib. Math. J. 46(2), 325–332 (2005)
Robinson, D.J.S.: A Course in the Theory of Groups, Volume 80 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (1996)
Tolue, B.: The non-cenralizer of a finite group. Math. Rep. 17(3), 265–275 (2015)
Tolue, B., Erfanian, A.: Relative non-commuting graph of a finite group. J. Algebra Appl. 12, 1250157 (2013). https://doi.org/10.1142/S0219498812501575
Yushmanov, S.V.: Estimate for the metric dimension of a graph in terms of the diameters and the number of vertices. Vestn. Mosk. Univ. Ser. I Mat. Mekh. 103, 69–70 (1987)
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Tolue, B. The twin non-commuting graph of a group. Rend. Circ. Mat. Palermo, II. Ser 69, 591–599 (2020). https://doi.org/10.1007/s12215-019-00421-4
- Non-commuting graph
- Twin graph
- Metric dimension
Mathematics Subject Classification
- MSC 20P05