پديد آورنده :
وارسته، مهدي
عنوان :
طيف فاصله ي گراف هاي كيلي گروه هاي كاكستر
مقطع تحصيلي :
كارشناسي ارشد
گرايش تحصيلي :
رياضي محض﴿جبر﴾
محل تحصيل :
اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي
صفحه شمار :
هشت،106ص.: جدول
يادداشت :
ص.ع.به فارسي وانگليسي
استاد راهنما :
بيژن طائري
استاد مشاور :
محمد مشكوري
تاريخ نمايه سازي :
15/10/92
استاد داور :
محمدرضا ريسمانچيان، قهرمان طاهريان
چكيده فارسي :
به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي
چكيده انگليسي :
The distance spectra of Cayley graphs of Coxeter groups Mehdi varasteh m varasteh@math iut ac ir September18 2013 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 Iran Supervisor Dr Bijan Taeri b taeri@cc iut ac ir Advisor Ghahreman Taherian G taherian@cc iut ac ir 2000 MSC 15A03 15A21 15A36 15A48 Key Words Cayley graph coxeter group Absolute order Weak order Spectrum reflections Abstract This Msc thesis is based on the following paper Rentlen P The distance spectra of Cayley graphs of Coxeter groups Discrete Mathematics 311 2011 738 755 Let W be a group and W S where S s1 sn si sj mij 1 1 i j n mij Z and mii 1 The group W is called a coxeter group and W S is called a coxeter system Let W S be a finite Coxeter system and let T wsw 1 w W s S be the set of all reflections of W Let X be a subset of generators of W satisfying X X 1 where X 1 x 1 x X and 1 X The cayley graph W X of W with respect to X is a graph with vertex set V W and edges set E w xw w W x X For the graphs associated to Coxeter groups the distance function has a well known group theoretic interpretation Every w W may be written as a word in the reflections T respectively simple reflections S and the minimum number of such reflections that must be used is the absolute length lT w respectively length lS w of w The length functions satisfy the properties l w l w 1 and l w 0 if and only if w 1 where henceforth l is generic for lS or lT Given lS respectively lT one defines two order relations on W called left and right weak respectively absolute order Right order on W is defined by u R v l u l u 1 v l v while left order is defined by u L v u 1 R v 1 We can define two natural classes of Cayley graphs on Coxeter groups the weak order graph W S the graph underlying the weak Bruhat order on W and the absolute order graph W T the graph underlying the absolute order on W Of particular interest are the Cayley graphs on the permutation group Gn W An 1 the weyl group of coxeter group of type An 1 Partial results are known for the Laplacian spectra of some of the weak order graphs and hence for their adjacency spectra Using the technique of equitable partitions Bacher was able to obtain a large class of eigenvalues of the Laplacian of Gn S Another natural graph invariant that has received less attention is the distance spectrum Let d u v be the length of a shortest path between u and v in The distance spectrum of is the spectrum of the matrix d whose u v entry is d u v The distance polynomial p q is the characteristic polynomial of d In this thesis we take the first steps toward the study of the distance spectra of the weak and absolute order graphs of the Coxeter groups We examine the calculation of these spectra exactly in a large number of cases which is unusual because the computation of distance spectra is generally considered more difficult than the computation of adjacency spectra In the absolute order case we observe that the eigenvalues are all integral Even more striking we find that in the weak order case there are generally only a few distinct eigenvalues suggesting the presence of additional underlying structures some features of which are revealed by our analysis The tools used to obtain the distance spectra of the weak and absolute order graphs have very different flavors attributable to the fact that the absolute length is a conjugacy class invariant This permits the use of simple character theoretic techniques in the absolute order case A wider variety of tools must be brought to bear in the weak order case yet one can obtain simple closed formulae for the distance spectra of the weak order graphs of types A D E and I and perhaps for many of the others We begin with a general discussion of the fundamental relationship between the distance matrices of interest here and the group algebras of the Coxeter
استاد راهنما :
بيژن طائري
استاد مشاور :
محمد مشكوري
استاد داور :
محمدرضا ريسمانچيان، قهرمان طاهريان