7116

6643

كارشناسي ارشد

رياضي محض﴿جبر﴾

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1390

[هشت]، 99ص.: مصور، جدول، نمودار

ص.ع. به فارسي و انگليسي

بيژن طائري

قهرمان طاهريان

1/8/91

عليرضا عبداللهي، عاطفه قرباني

1396/09/20

كتابنامه

علوم رياضي

رياضي

ID6643

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

The spectrum of semi Cayley graphs over abelian groups MAJID AHMADPOUR m Ahmadpour@math iut ac ir January 28 2012 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr Bijan Taeri b taeri@cc iut ac irAdvisor Ghahreman Taherian G taherian@cc iut ac ir2000 MSC 15A03 15A21 15A36 15A48Key Words Cayley graph Semi Cayley graph Integral graph automorphism group Spectrum Semi regular and regular actionAbstract This Msc thesis is based on the following paperXing Gao Yanfeng Luo The spectrum of semi Cayley graphs over abelian groups Linear Algebraand its Applications 432 2010 2974 2983 Let G be a group and H be a subset of G such that e H where e is the identity element of G The Cayley digraph Cay G H over G with respect to H is a graph with vertex set V G and edgeset E x y x y G yx 1 H If H 1 H then Cay G H is an undirected graph and it iscalled a Cayley graph Equivalently a Cayley graph may be de ned as a graph V E which admitsan automorphism group acting regularly on the vertex set V Let R S T be subsets of a group G suchthat R R 1 S S 1 and e R S De ne the undirected graph SC G R S T to have vertex set G 0 1 and with vertices h i g j adjacent if and only if one of the following three possibilities occurs 1 i j 0 and gh 1 R 2 i j 1 and gh 1 S 3 i 0 j 1 and gh 1 T The graph SC G R S T is called a semi Cayley graph Equivalently a semi Cayley graph may bede ned as a graph V E which admits an automorphism group acting semiregular on the vertex setV with two orbits of equal size Let be a graph with vertices labeled as 0 1 n 1 The adjacencymatrix A of is an n n matrix with i j entry equals to 1 if vertices i and j are adjacent and0 otherwise The spectrum of a graph is the set of numbers which are eigenvalues of A togetherwith their multiplicities We shall usually refer to the eigenvalues of A as the eigenvalues of Theaim of this thesis is to study the spectrum of semi Cayley graphs over nite abelian groups We derive aformula of the spectrum of semi Cayley graphs over nite abelian groups As an application of our mainresult we give a method to construct integral graphs In particular we obtain an explicit expression forthe spectrum of Cayley graphs overtwo non abelian groups dihedral groups and dicyclic groups Let SC G R S T be a semi Cayley graph over a nite abelian group G Zn1 Znt we provethat has eigenvalues R rt S1 rt R rt S1 rt 2 4 T1 rt 2 r1 r r1 r r rj 0 1 nj 1 j 1 2 t 2rj 0 1 nj 1 j 1 2 t where R rt S1 rt and T1 rt are the eigenvalues of Cay G R r1 r rCay G S and Cay G T respectively A graph is called an integral graph if it has an integral spectrum i e all eigenvalues are integers Let SC G R S T be a semi Cayley graph over G Zn1 Znt If Cay G R and Cay G T areintegral graphs we show that is an integral graph We give the spectrum of Cayley graphs over dihedralgroups Dn and dicyclic groups DC2n respectively Let Dn a x an x2 1 x 1 ax a 1 be thedihedral group H Dn e with H 1 H and H i ai H Denote by 1 and 2 the subgraphsinduced by ai 0 i n 1 and ai x 0 i n 1 in Cay Dn H respectively Let Cay Dn H be a Cayley graph Then we show that 1 If H ai x 0 i n 1 then Cay Dn H has eigenvalues ai H n with multiplicity 2 irr 0 1 n 1 2 If H ai x 0 i n 1 let ai0 x H then Cay Dn H has eigenvalues ir ai H n i i r n 0 r 0 1 n 1 ai x H

بيژن طائري

قهرمان طاهريان

عليرضا عبداللهي، عاطفه قرباني