Rajkumar, R.; Devi, P. Permutability graphs of subgroups of some finite non-abelian groups. (English) Zbl 1346.05116 Discrete Math. Algorithms Appl. 8, No. 3, Article ID 1650047, 26 p. (2016). Summary: In this paper, we study the structure of the permutability graphs of subgroups, and the permutability graphs of non-normal subgroups of the following groups: the dihedral groups \(D_n\), the generalized quaternion groups \(Q_n\), the quasi-dihedral groups \(QD_{2^n}\) and the modular groups \(M_{p^n}\). Further, we investigate the number of edges, degrees of the vertices, independence number, dominating number, clique number, chromatic number, weakly perfectness, Eulerianness, Hamiltonicity of these graphs. Cited in 1 ReviewCited in 6 Documents MSC: 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C07 Vertex degrees 05C15 Coloring of graphs and hypergraphs 05C45 Eulerian and Hamiltonian graphs 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.) 20K27 Subgroups of abelian groups Keywords:permutability graphs; non-abelian groups; independence number; dominating number; weakly perfect; Eulerian; Hamiltonian PDFBibTeX XMLCite \textit{R. Rajkumar} and \textit{P. Devi}, Discrete Math. Algorithms Appl. 8, No. 3, Article ID 1650047, 26 p. (2016; Zbl 1346.05116) Full Text: DOI arXiv References: [1] 1. A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algebra28 (2006) 468-492. genRefLink(16, ’S1793830916500476BIB001’, ’10.1016 · Zbl 1105.20016 [2] 2. A. Abdollahi and A. Mohammadi Hassanabadi, Non-cyclic graph of a group, Comm. Algebra35 (2007) 2057-2081. genRefLink(16, ’S1793830916500476BIB002’, ’10.1080 · Zbl 1131.20016 [3] 3. E. A. Bertram, M. Herzog and A. Mann, On a graph related to conjugacy classes of groups, Bull. Lond. Math. Soc.22 (1990) 569-575. genRefLink(16, ’S1793830916500476BIB003’, ’10.1112 · Zbl 0743.20017 [4] 4. M. Bianchi, A. Gillio and L. Verardi, Finite groups and subgroup-permutability, Ann. Mat. Pura Appl.169 (1995) 251-268. genRefLink(16, ’S1793830916500476BIB004’, ’10.1007 · Zbl 0849.20014 [5] 5. M. Bianchi, A. Gillio and L. Verardi, Subgroup-permutability and affine planes, Geom. Dedicata85 (2001) 147-155. genRefLink(16, ’S1793830916500476BIB005’, ’10.1023 · Zbl 0987.20009 [6] 6. J. P. Bohanon and L. Reid, Finite groups with planar subgroup lattices, J. Algebra Comb.23 (2006) 207-223. genRefLink(16, ’S1793830916500476BIB006’, ’10.1007 · Zbl 1133.20013 [7] 7. A. Gillio and L. Verardi, On finite groups with a reducible permutability-graph, Ann. Mat. Pura Appl.171 (1996) 275-291. genRefLink(16, ’S1793830916500476BIB007’, ’10.1007 · Zbl 0872.20021 [8] 8. M. Tǎrnǎuceanu, Subgroup commutativity degrees of finite groups, J. Algebra321 (2009) 2508-2520. genRefLink(16, ’S1793830916500476BIB008’, ’10.1016 [9] 9. M. J. Erickson and A. Vazzana, Introduction to Number Theory (Chapman and Hall/CRC, Florida, 2008). [10] 10. R. Rajkumar and P. Devi, Planarity of permutability graphs of subgroups of groups, J. Algebra Appl.13 (2014) 15; Article No. 1350112. [Abstract] · Zbl 1287.05060 [11] 11. R. Rajkumar and P. Devi, On permutability graphs of subgroups of groups, Discrete Math. Algorithms Appl.07 (2015) 11; Article No. 1550012. [Abstract] · Zbl 1316.05062 [12] 12. D. J. S. Robinson, A Course in the Theory of Groups (Springer-Verlag, New York, 1996). genRefLink(16, ’S1793830916500476BIB012’, ’10.1007 [13] 13. J. S. Williams, Prime graph components of finite groups, J. Algebra69 (1981) 487-513. genRefLink(16, ’S1793830916500476BIB013’, ’10.1016 [14] 14. D. B. West, Introduction to graph theory (Prentice-Hall, New Delhi, 2006). This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.