relative n-th non-commuting graphs of finite groups

‎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 $x^{n}yeq yx^{n}$‎. ‎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.