THE GENERALISED NON-COMMUTING GRAPH OF A FINITE GROUP

In this paper we define the generalised non-commuting graph Gamma((H,K)), where H and K are two subgroups of a non-abelian group G. Take (H boolean OR K) \ (C-H (K) boolean OR C-K (H)) as the vertices of the graph and two distinct vertices x and y join, whenever x or y is in H and [x, y] not equal 1. We obtain diameter and girth of this graph. Also, we discuss the dominating set and planarity of Gamma((H,K)). Moreover, we try to find a connection between Gamma((H,K)) and the relative commutativity degree of two subgroups d(H, K). Furthermore, we prove that if Gamma((H,G)) congruent to Gamma((K,G)), then Gamma(H) congruent to Gamma(K). And finally we introduce a special case when subgroup K is equal to the non-abelian group G.