Generalization of non-commuting graph of a finite group

Abstract

In this paper we define the generalized non-commuting graph ?(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] 6= 1. We obtain diameter and girth of this graph. Also, we discuss the dominating set and planarity of ?(H,K,L). Moreover, we try to find a connection between ?(H,K,L) and the relative commutativity degree of three subgroups d(H, K ?L).