The essence of the non-commuting graph remind us to find a connection between this graph and the commutativity degree as denoted by d(G). On the other hand, d(H, G) the relative commutativity degree, was the key to generalize the non-commuting graph ΓG to the relative non-commuting graph (denoted by ΓH, G) for a non-abelian group G and a subgroup H of G. In this paper, we give some results about ΓH, G which are mostly new. Furthermore, we prove that if (H1, G1) and (H2, G2) are relative isoclinic then ΓH1, G1 ≅ Γ H2, G2 under special conditions.

In this paper we introduce the conjugate graph [Formula: see text] associated to a nonabelian group G with vertex set G\Z(G) such that two distinct vertices join by an edge if they are conjugate. We show if [Formula: see text], where S is a finite nonabelian simple group which satisfy Thompson's conjecture, then G ≅ S. Further, if central factors of two nonabelian groups H and G are isomorphic and |Z(G)| = |Z(H)|, then H and G have isomorphic conjugate graphs.

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.

This article has been prepared based on a new concept of an ultra-group H M which is depend on a group G and its subgroup H. Our aim is to introduce the category of ultra-groups and investigate about some properties of this category.

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.