Let G be a finite group and N be a normal subgroup of G. We define an undirected simple graph Γ N ,G to be a graph whose vertex set is all elements in G \ Z N (G) and two vertices x and y are adjacent ifffor all x ∈ G}. If N = 1, then we obtain the known non-commuting graph of G. We give some basic results about connectivity, regularity, planarity, 1-planarity and some numerical invariants of the graph which are mostly improvements of the results given for non-commuting graphs. Also, a probability related to this graph is defined and a formula for the number of edges of the graph in terms of this probability is given.