On the complement of the intersection graph of subgroups of a group

Document Type : Original paper

Authors

1 Department of Mathematics, Sri Paramakalyani College, Alwarkurichi - 627 412, Tamil Nadu, India.

2 Department of Mathematics, The Gandhigram Rural Institute (Deemed to be University), Gandhigram - 624 302, Tamil Nadu, India

Abstract

The complement of the intersection graph of subgroups of a group G, denoted by Ic(G), is the graph whose vertex set is the set of all nontrivial proper subgroups of G and its two distinct vertices H and K are adjacent if and only if H∩K is trivial. In this paper, we classify all finite groups whose complement of the intersection graph of subgroups is one of totally disconnected, bipartite, complete bipartite, tree, star graph or C3-free. Also we characterize all the finite groups whose complement of the intersection graph of subgroups is planar.

Keywords

Main Subjects

References

[1] H. Ahmadi and B. Taeri, Planarity of the intersection graph of subgroups of a finite group, J. Algebra Appl. 15 (2016), no. 3, Article ID: 1650040.
https://doi.org/10.1142/S0219498816500407
[2] S. Akbari, F. Heydari, and M. Maghasedi, The intersection graph of a group, J. Algebra Appl. 14 (2015), no. 5, Article ID: 1550065.
https://doi.org/10.1142/S0219498815500656
[3] J. Bosák, The graphs of semigroups, in: Theory of Graphs and Application, Academic Press, New York (1964), 119–125.
[4] W. Burnside, Theory of Groups of Finite Order, Dover Publications, Cambridge, 1955.
[5] B. Csákány and G. Pollák, The graph of subgroups of a finite group, Czechoslovak Math. J. 19 (1969), no. 94, 241–247.
http://dx.doi.org/10.21136/CMJ.1969.100891
[6] P. Devi, Studies on graphs associated with subgroups of groups, Ph.D. thesis, Gandhigram, India, 2016.
[7] P. Devi and R. Rajkumar, Intersection graph of abelian subgroups of a group, Indian J. Discrete Math. 2 (2016), no. 2, 77–86.
[8] F. Harary, Graph Theory, Addison-Wesley, Philippines, 1969.
[9] S.H. Jafari and N. Jafari Rad, Intersection graphs of normal subgroups of groups, Quasigroups and related systems 18 (2010), no. 2, 137–142.
[10] S. Kayacan, Connectivity of intersection graphs of finite groups, Comm. Algebra 46 (2018), no. 4, 1492–1505.
https://doi.org/10.1080/00927872.2017.1347662
[11] H.L. Lin, On groups of order p2q, p2q2, Tamkang J. Math. 5 (1974), no. 2, 167–190.
[12] X. Ma, On the diameter of the intersection graph of a finite simple group, Czechoslovak Math. J. 66 (2016), no. 2, 365–370.
https://doi.org/10.1007/s10587-016-0261-2
[13] R. Rajkumar and P. Devi, Toroidality and projective-planarity of intersection graphs of subgroups of finite groups, arXiv preprint arXiv:1505.08094 (2015).
[14] R. Rajkumar and P. Devi, Intersection graph of subgroups of some non-abelian groups, Malaya J. Mat. 4 (2016), no. 02, 238–242.
[15] R. Rajkumar and P. Devi, Intersection graphs of cyclic subgroups of groups, Electron. Notes Discrete Math. 53 (2016), 15–24.
https://doi.org/10.1016/j.endm.2016.05.003
[16] H. Shahsavari and B. Khosravi, Characterization of some families of simple groups by their intersection graphs, Comm. Algebra 48 (2020), no. 3, 1266–1280.
https://doi.org/10.1080/00927872.2019.1682151
[17] R. Shen, Intersection graphs of subgroups of finite groups, Czechoslovak Math. J. 60 (2010), no. 4, 945–950.
https://doi.org/10.1007/s10587-010-0085-4
[18] S. Visweswaran and P. Vadhel, Some results on the complement of the intersection graph of subgroups of a finite group, J. Algebraic Sys. 7 (2020), no. 2, 105–130.
https://doi.org/10.22044/jas.2018.5917.1296
[19] B. Zelinka, Intersection graphs of finite abelian groups, Czechoslovak Math. J. 25 (1975), no. 2, 171–174.
https://doi.org/10.21136/CMJ.1975.101307
 
Communications in Combinatorics and Optimization
Volume 10, Issue 1
March 2025
Pages 57-68

Files

Share

How to cite

Statistics