The distance spectra of Cayley graphs of Coxeter groups Mehdi varasteh rn.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.taerj®cc. jut .ac. jr Advisor: Ghahreman Taherian, G . taherjancc. jut . ac . jr 2000 MSC: 15A03 15A2l l5A36 15A48 Key Words: Cayley graph, coxeter group,Absolute order, Weak order, Spectrum, reflections : 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 = {s 1 ,. . s n | (sisj) mij = 1, 1 ? i,j ? n},m ij e Z and m ij = 1. The group W is called a coxeter group and (W, ) is called a coxeter system. Let (W,S) be a finite Coxeter system and let T= {wsw -1 : w e W , s e 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 | c e X} and 1 I X. The cayley graph G(W, X) of W with respect to X is a graph with vertex set V(G) = W and edges set E(G) = {{w, xw} : w e W x e X}. For the graphs associated to Coxeter groups, the distance function has a well known group theoretic interpretation. Every w e 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 l T (w) (respectively, length 1 s (w)) of w. The length functions satisfy the properties 1(w) = 1 (w -1 ), and 1(w) = 0 if and only if w = 1, where henceforth 1 is generic for l s or 1 T . Given l s (respectively l T ) one defines two order relations on W, called left and right weak (respectively, absolute) order. Right order on W is defined by while left order is defined by u ? L u « U -1 R u -1 . We can define two natural W(A n-1 ), the weyl group of coxeter group of type A_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 partially answered along the way.
