Suppose $n$ is a fixed positive integer. We introduce the relative n-th non-commuting graph $Gamma^{n} _{H,G}$, associated to the non-abelian subgroup $H$ of group $G$. The vertex set is $Gsetminus C^n_{H,G}$ in which $C^n_{H,G} = {xin G : [x,y^{n}]=1 mbox{~and~} [x^{n},y]=1mbox{~for~all~} yin H}$. Moreover, ${x,y}$ is an edge if $x$ or $y$ belong to $H$ and $xy^{n} eq y^{n}x$ or $x^{n}y eq yx^{n}$. In fact, the relative n-th commutativity degree, $P_{n}(H,G)$ the probability that n-th power of an element of the subgroup $H$ commutes with another random element of the group $G$ and the non-commuting graph were the keys to construct such a graph. It is proved that two isoclinic non-abelian groups have isomorphic graphs under special conditions.
Erfanian, A., & Tolue, B. (2013). Relative n-th non-commuting graphs of finite groups. Bulletin of the Iranian Mathematical Society, 39(4), 663-674.
MLA
A. Erfanian; B. Tolue. "Relative n-th non-commuting graphs of finite groups". Bulletin of the Iranian Mathematical Society, 39, 4, 2013, 663-674.
HARVARD
Erfanian, A., Tolue, B. (2013). 'Relative n-th non-commuting graphs of finite groups', Bulletin of the Iranian Mathematical Society, 39(4), pp. 663-674.
VANCOUVER
Erfanian, A., Tolue, B. Relative n-th non-commuting graphs of finite groups. Bulletin of the Iranian Mathematical Society, 2013; 39(4): 663-674.