Abstract. In this paper we introduce the non-coprime graph associated to the group G with vertex set } { \ e G such that two distinct vertices are adjacent whenever their orders are relatively non-coprime. Some numerical invariants like diameter, girth, dominating number, independence and chromatic numbers are determined and it has been proved that the non-coprime graph associated to a group G is planar if and only if G is isomorphic to one of the groups 6 5 2 2 4 3 2 , , , , , ' ' ' ' ' ' ' u or 3 S . Moreover, we prove that non-coprime graph of a nilpotent group G is regular if and only if G is a p -group, where p is prime number. Furthermore, a connection between the non- coprime graph and known prime graph has been stated here.

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