On groups with specified quotient power graphs

Document Type: Research Paper

Authors

Yazd University

Abstract

In this paper we study some relations between the power and quotient power graph of a finite group‎. ‎These interesting relations motivate us to find some graph theoretical properties of the quotient‎ ‎power graph and the proper quotient power graph of a finite group $G$‎. ‎In addition‎, ‎we classify those groups whose quotient (proper quotient) power graphs are isomorphic to trees or paths‎.

Keywords

Main Subjects


[1] J. Abawa jy, A. Kelarev and M. Chowdhury, Power graphs: A Survey, Electron. J. Graph Theory Appl. (EJGTA), 1 (2013) 125-147.

[2] A. Ab dollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algebra, 298 (2006) 468-492.

[3] D. Bubb oloni, S. Dol , M. A. Iranmanesh and C. E. Praeger, On bipartite divisor graphs for group conjugacy class sizes, J. Pure Appl. Algebra, 213 (2009) 1722-1734.

[4] D. Bubb oloni, M. A. Iranmanesh and S. M. Shaker, Quotient graphs for power graphs, Submitted.

[5] D. Bubb oloni, M. A. Iranmanesh and S. M. Shaker, 2-connectivity of the power graph of nite alternating groups, Submitted 2014, arXiv preprint arXiv:1412.7324.

[6] P. J. Cameron, The power graph of a nite group II, J. Group Theory, 13 (2010) 779-783.

[7] P. J. Cameron and S. Ghosh, The power graph of a nite Group, Discrete Math., 311 (2011) 1220-1222.

[8] I. Chakrabarty, S. Ghosh and M. K. Sen, Undirected p ower graphs of semigroups, Semigroup Forum, 78 (2009) 410-426.

[9] M. Deaconescu, Classi cation of nite groups with all elements of prime order, Proc. Amer. Math. Soc., 106 (1989) 625-629.

[10] A. Do ostabadi, A. Erfanian and A. Jafarzadeh, Some results on the power graph of groups, The Extended Abstracts of the 44th Annual Iranian Mathematics Conference 27-30 August 2013, Ferdowsi University of Mashhad, Iran.

[11] M. A. Iranmanesh and C. E. Praeger, Bipartite divisor graphs for integer subsets, Graphs Combin., 26 (2010) 95-105.

[12] A. V. Kelarev, Graph algebras and automata, Marcel Dekker, New York, 2003.

[13] A. V. Kelarev, Lab elled Cayley graphs and minimal automata, Australas. J. Combin., 30 (2004) 95-101.

[14] A. V. Kelarev and S. J. Quinn, A combinatorial prop erty and power graphs of groups, Contrib. General Algebra, 12 (2000) 229-235.

[15] A. V. Kelarev, S. J. Quinn and R. Smolikova, Power graphs and semigroups of matrices, Bul l. Austral. Math. Soc., 63 (2001) 341-344.

[16] A. V. Kelarev and S. J. Quinn, Directed graphs and combinatorial prop erties of semigroups, J. Algebra, 251 (2002) 16-26.

[17] A. V. Kelarev and S. J. Quinn, A combinatorial prop erty and p ower graphs of semigroups, Comment. Math. Univ. Carolin., 45 (2004) 1-7.

[18] A. V. Kelarev, J. Ryan and J. Yearwo o d, Cayley graphs as classi ers for data mining: The in
uence of asymmetries, Discrete Math., 309 (2009) 5360-5369.

[19] M. Mirzargar, A. R. Ashra and M. J. Nadja -Arani, On the p ower graph of a nite group, Filomat, 26 (2012)
1201-1208.

[20] G. R. Pourgholi, H. Youse -Azari and A. R. Ashra , The undirected p ower graph of a nite group, Bul l. Malays. Math. Sci. Soc., 38 (2015) 1517-1525.

[21] J. S. Rose, A course on group theory, Dover Publications, Inc., New York, 1994.

[22] D. B. West, Introduction to graph theory, Prentice Hall. Inc. Upp er Saddle River, NJ, 1996.