Relative g-noncommuting graph of finite groups

Let G be a finite group. For fixed element g in and given subgroup H of , the relative -noncommuting graph is simple undirected whose vertex set two vertices x y are adjacent if ∈ or [ ]≠ −1 . We denote this by Γ In paper, we obtain computing formulae for degree any characterize whether tree, star graph, lollipop complete together with some properties involving isomorphism graphs. also present certain relations between number edges generalized commuting probabilities which give Finally, conclude paper deriving bounds