The orbit graph of finite non-Abelian groups

Abstract

Let G be a finite non-abelian group and let Ω be a set of elements of G. Let A be the set of commuting elements in Ω, i.e A = {v ∈ Ω : vg = gv, g ∈ G}. In this paper, we extend the work on conjugate graph by defining a new graph called the orbit graph, denoted as ΓGΩ. The vertices of ΓGΩ are non central elements in Ω but not in A in which two vertices of ΓGΩ are adjacent if they are conjugate. Some graph properties are provided. Besides, the orbit graph of dihedral groups and quaternion groups is determined.