Let G be a ﬁnite group and H,K be two subgroups of G. We introduce the relative non-normal graph of K with respect to H , denoted by NH,K, which is a bipartite graph with vertex sets HHK and KNK(H) and two vertices x ∈ H HK and y ∈ K NK(H) are adjacent if xy / ∈ H, where HK =Tk∈K Hk and NK(H) = {k ∈ K : Hk = H}. We determined some numerical invariants and state that when this graph is planar or outerplanar.

Type of Study: Research paper |
Subject:
Special

1. A. Abdollahi, Engel graph associated with a group, J. Algebra, 318 (2007), 680-691. [DOI:10.1016/j.jalgebra.2007.09.007]

2. A. Abdollahi and A. Mohammadi Hassanabadi, Non-cyclic graph of a group, Comm. in Algebra, 35 (2007), 2057-2081. [DOI:10.1080/00927870701302081]

3. A. Abdollahi, S. Akbari and H. R. Maimani, Non-commuting graph of a group, J. Algebra, 298 (2006), 468-492. [DOI:10.1016/j.jalgebra.2006.02.015]

4. M. Bodirsky, O. Gimenez, M. Kang and M. Noy, Enumeration and limit laws of seriesparallel graphs, European Journal of Combinatorics, 28, (2005), 2091-2105. [DOI:10.1016/j.ejc.2007.04.011]

5. J. A. Bondy and J. S. R. Murty, Graph Theory with Applications, Elsevier, (1977). [DOI:10.1007/978-1-349-03521-2]

6. G. Chartrand and P. Zhang, Chromatic Graph Theory Taylor & Francis, 2009. [DOI:10.1201/9781584888017]

7. A. Erfanian, M. Farrokhi D.G., and B. Tolue, Non-normal graphs of finite groups, J. Algebra Appl, 12 (2013) [9 pages] Doi:10.1142/S0219498812501939. [DOI:10.1142/S0219498812501939]

8. F. Saeedi, M. Farrokhi D. G. and S. H. Jafari, Subgroup normality degrees of finite groups I, Arch. Math., 96 (2011), 215-224. [DOI:10.1007/s00013-011-0234-5]