Relative n-th non-commuting graphs of finite groups

Suppose nn is a fixed positive integer. We introduce the relative n-th non-commuting graph GammanH,GGammaH,Gn, associated to the non-abelian subgroup HH of group GG. The vertex set is GsetminusCnH,GGsetminusCH,Gn in which CnH,G=xinG:[x,yn]=1mbox and [xn,y]=1mbox for all yinHCH,Gn=xinG:[x,yn]=1mbox and [xn,y]=1mbox for all yinH. Moreover, x,yx,y is an edge if xx or yy belong to HH and xyneqynxxyneqynx or xnyeqyxnxnyeqyxn. In fact, the relative n-th commutativity degree, Pn(H,G)Pn(H,G) the probability that n-th power of an element of the subgroup HH commutes with another random element of the group GG 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.