GENERALIZATION OF NON-COMMUTING GRAPH OF A FINITE GROUP

Abstract

In this paper we define the generalized non-commuting graph Γ(H,K,L) where H, K and L are three subgroups of a non-abelian group G. Take (H ∪K ∪L)\CH (K ∪L)∪ CK∪L(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] ̸= 1. We obtain diameter and girth of this graph. Also, we discuss the dominating set and planarity of Γ(H,K,L). Moreover, we try to find a connection between Γ(H,K,L) and the relative commutativity degree of three subgroups d(H, K ∪ L).