The aim of this paper is to give some results on the graph associated to a finite group G and subset S of G – Z(G) as defined in [9] with vertex set consisting are all elements in G – (Z(G) ∪ S). Two distinct vertices x and y are adjacent if and only there exists an element s ∈ S such that [x, s] = [y, s] = 1, where [x, s] = x–1 s–1 xs is the commutator of elements x and s. The results stated here are mostly new or an improvement of results give in the above paper.