10491

784 دكتري

دكتري

رياضي

اصفهان: دانشگاه صنعتي اصفهان ، دانشكده علوم رياضي

1394

هفت، 119ص.: مصور، جدول، نمودار

بيژن طائري

عليرضا اشرفي

فارسي به انگليسي

1394/07/12

غلامرضا رضايي زاده، مريم خاتمي بيدگلي، غلامرضا اميدي اردلي

رياضي

ID784 دكتري

به فارسي و انگليسي

n Cayley graphs over nite groups Majid Arezoomand arezoomand@math iut ac ir August 19 2015 Doctor of Philosophy Thesis Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Bijan Taeri b taeri@cc iut ac ir Advisor Dr Alireza Ashra ashra @kashanu ac ir Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Department of Pure Mathematics Faculty of Mathematical Sciences University of Kashan Kashan 87317 51167 Iran Abstract A digraph is called n Cayley digraph over a group G if there exists a semiregular subgroup RG of Aut isomorphic to G with n orbits In this thesis we decompose the characteristic polynomial of to the characteristic polynomial of smaller matrices in terms of irreducible representations of G Let S be a subset of a group G BCay G S is a special class of 2 Cayley graphs which is called a bi Cayley graph Its vertex set is G 1 2 and its edge set is g 1 sg 2 g G s S A nite group G is called a bi Cayley integral group if for any subset S of G BCay G S is a graph with integer eigenvalues integral graph We classify all bi Cayley integral groups and all nite groups admitting a connected integral bi Cayley graph of valency k k 2 3 Also we determine all nite abelian groups admitting a connected quartic integral bi Cayley graph Let G be a nite group An element g G is called non vanishing if for every irreducible complex character of G g 0 Using the eigenvalues of bi Cayley graphs we prove that if G p where p is the smallest prime divisor of G then non vanishing elements of G are central elements Also we prove that if Cl g 2 3 and o g 6 1 then g is a non vanishing element A bi Cayley graph BCay G S is called a BCI graph if BCay G S BCay G T implies for some g G and Aut G A group G is called a BCI group if all that T gS bi Cayley graphs of G are BCI graphs We prove a Babai type theorem for BCI graphs We prove that Zp p is a prime is a BCI group Also we prove that every Sylow p subgroup p 3 of a BCI group is elementary abelian Furthermore we prove that every BCI group is a solvable group Let be a 2 Cayley digraph over G We say that is normal if RG is a normal subgroup of Aut In this thesis we determine the normalizer of RG in Aut We show that the automorphism group of each normal 2 Cayley digraph over a group with solvable automor phism group is solvable We prove that for each nite group G Q8 Zr r 0 where Q8 2 is the quaternion group of order 8 and Z2 is the cyclic group of order 2 there exists a normal 2 Cayley graph over G and that every nite group has a normal 2 Cayley digraph Also we determine all non normal 2 Cayley graphs over a group of prime order Keywords n Cayley graph semi Cayley graph bi Cayley graph group representation graph isomorphism integral graph normal 2 Cayley graph MSC 2010 05C25 05C31 05C50 05C60 05E10 20C10 20C15 20B25 20D10 1

بيژن طائري

عليرضا اشرفي

غلامرضا رضايي زاده، مريم خاتمي بيدگلي، غلامرضا اميدي اردلي