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 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 noncoprime graph and known prime graph has been stated here.

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.

Let be a nite group. In this paper we introduce the generalized conjugate graph Γ ( , ) which is a graph whose vertices are all the non-central subsets of with elements and two distinct vertices and joined by an edge if = for some ∈ . General properties of the graph such as the number of edges, clique, chromatic, dominating, independence numbers, automorphism and energy of the graph are discussed. We also present a condition under which two generalized conjugate graphs are isomorphic. Moreover, the generalized conjugate graph is a key to de ne the probability that two subsets of the group with the same cardinality are conjugate. We obtain some upper and lower bounds for this probability.