International Journal of Pure and Applied Mathematics Volume 102 No. 4 2015, 747-755 ISSN: 1311-8080 (printed version); ISSN: 1314-3395 (on-line version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v102i4.12 P A ijpam.eu THE ORBIT GRAPH OF FINITE NON-ABELIAN GROUPS S.M.S. Omer 1 § , N.H. Sarmin 2 , A. Erfanian 3 1 Department of Mathematics Faculty of Science University of Benghazi Benghazi, Libya 2 Department of Mathematical Sciences Faculty of Science Universiti Teknologi Malaysia Johor Bahru, MALAYSIA 3 Department of Mathematics and Center of Excellence in Analysis on Algebraic Structures Ferdowsi University of Mashhad Mashhad, IRAN 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 Γ Ω G . The vertices of Γ Ω G are non central elements in Ω but not in A in which two vertices of Γ Ω G are adjacent if they are conjugate. Some graph properties are provided. Besides, the orbit graph of dihedral groups and quaternion groups is determined. AMS Subject Classification: 20P05, 20B40, 97K30 Key Words: commutativity degree, graph theory, group action, conjugate graph Received: March 29, 2015 c  2015 Academic Publications, Ltd. url: www.acadpubl.eu § Correspondence author