ABSTRACT: Let G be a finite group and N be a normal subgroup of G. We define an undirected simple graph N,G to be a graph whose vertex set is all elements in G n ZN (G) and two vertices x and y are adjacent iff [x, y] = 2 N, where ZN (G) = fg 2 G : [x, g] 2 N for all x 2 Gg. 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.

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