IJPAM: Volume 102, No. 4 (2015)

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

Received: March 29, 2015

AMS Subject Classification: 20P05, 20B40, 97K30

Key Words and Phrases: commutativity degree, graph theory, group action, conjugate graph

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DOI: 10.12732/ijpam.v102i4.12 How to cite this paper?

Source:
International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2015
Volume: 102
Issue: 4
Pages: 747 - 755


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