Research Journal of Recent Sciences _________________________________________________ ISSN 2277-2502 Vol. 4(6), 142-143, June (2015) Res.J.Recent Sci. International Science Congress Association 142 Short Communication The Study of the Number of 2-Matchings in Graphs Ebrahimi S.H. and Khani M.HDepartment of mathematics, College of Basic Sciences, Hendijan Branch, Islamic Azad University, Hendijan, IRAN Department of Mathematics, College of Basic Sciences, Shahinshar Branch, Islamic Azad University, Shahinshahr, IRAN Available online at: www.isca.in,www.isca.me Received 2nd February 2014, revised 14th April 2014, accepted 13th July 2014Abstract In this paper we study the 2-matching of finite groups. We use conjugation and character of groups to calculate the number of 2-matchings for some import groups and using them we prove that a finite group G is abelian if and only if the number of its 2-matching is zero. Keywords: Matching, conjugation, character, non commuting graph. IntroductionLet be a non abelian group and  the center of . The related graphof is a graph in which the set of its vertices is \n and two vertices \rare connected if \r\r . in another word  \r \r. Study the groups using the properties of their graphs is a very important field of study in group theory1,2,. The theorems used in graph theory can be found in Solmon R., All Simple Groups are Characterized by their Graphs. If is a graph a matching for is a global subgraph in which components are vertices and edges. A, 2-matching is a matching with two edges. The number of 2- matching is shown by   A two matching for a group is a 2-matching for the related graph so    . For a graph , the  is the diameter of and the circumference is shown by  ! If is an arbitrary member of group , the degree of in is shown by As a direct consequence \n"  In which   is the centralizer of in  For more results about the prime group of a graph4,5,6 and applications of group theory are given in several titurature7-11. Matching of graphs To prove the main theorem of the paper we need some prior propositions: Proposition1: if is a finite group then  # and is a connected graph12. Proposition2:  ! $ in which is a non abelian and finite group12. Proposition3: if is a non abelian and infinite then is Hamiltonian, Proof: see refrence12. In the following proposition we obtain a formula that gives us the number of 2- matching’s in a graph. This formula is quite simple and based of the centralizer of those members that are not in the center of group. Proposition:     % &\n'()*+\n-.\n' ()*+Proof: according to Usman M. et. al.10 if is a simple graph and  0 the number of vertices and edges and the degree of vertex in then  23\n-4178But &\n' ()*+\nand for an element we know, \n' so the result follows. Example: 1: Consider the group:  :    $ $  $  $ Then:   ?  = $   $= $   $  $    $  $ = $     $  $ So the representation of the is figure-1. Example 2  We calculate this using the conjugation class of the group. The conjugation class of is shown in table-1. Research Journal of Recent Sciences ______________________________________________________________ISSN 2277-2502Vol. 4(6), 142-143, June (2015) Res.J.Recent Sci International Science Congress Association 143 Figure-1 The representation graph of Table-1 The Conjugation class of 1 12 3 x 6 2 3 ( ) xCS4 1 3 2 ( ) Class So according to the table  F9; G therefor by the proposition 2:  45\n45\n$45 Example 3  IIJ, The conjugation class of is shown in table-2. Table-2 The conjugation class of 1 2 12 2 2 13 4 x 24 4 8 3 4 ( ) xCS4 1 6 3 8 6 ( ) Class So  9MFN98F;8LFL9M J therefor by proposition2:  IIJWe can also similarly prove the following:  PIQ   QGPQIWe know that if is an abelian group then   # here we prove a theorem that shows the converse is true. Theorem: if   # then is abelian. Proof: by contrary let is non abelian then according to proposition 3,  ! $ and so S$. Also by propositions (1) and (3) , is a connected Hamilitonian graph. Now if HThen T# that is a contradiction. So  $consequently \n $ and we have:  $ so  $ this means U\n$ so U VH If U  then  $and so Qand by assumption is non abelian so W: but according to proposition (3),      that is a contradiction. If U H then   and then H therefor is abelian that is a contradiction as well. So is abelian. Conclusion The study of groups using methods of graph theory is widely used in group theory. One of this tools is the number of matching in graphs. The result of this paper show that a combination of matching with some other concepts in group theory such as conjugation give important properties of groups as we saw the number of 2-matchings somehow shows if group is abelian or not. So it will be a key for further works to use matching’s to figure out other properties of finite groups. References 1.Williams J., Prime Graph Component of Finite Groups, J. Algebra., 69(2), 487-513 (1981)2.Moghadamfar A.R., Shi WJ, Zhou W and Zokayi AR, On the Noncommuting graph Associated with a Finite Group, Siberian mathematical journal, 46(2), 325-332 (2005)3.Solmon R. and Wolder A., All Simple Groups are Characterized by their Graphs, Journal of group theory, 16(6), 893-984 (2013)4.Darafsheh M.R., Bigdly H. and Bahrami A., Some Results on non Commuting Graph of a Finite Group, Italian J.P.A. Mth., , 107-118 (2010) 5.Bondy J. A., Murty J. S. 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