The Terwilliger Algebraic Structures Of Halved Hypercube And Halved Folded Hypercube

Let ? =(X,R)denote a finite connected bipartite graph with diameter at least 3.Define the graph ? 2 as follows.The vertex set is X and two vertices x,y are adjacent whenever(?)(x,y)= i in ?.It is well known that ?2 has two connected components.The induced subgraph of ?2 on each connected component is called the halved graph of?,denoted by 1/2?.Let 1/2H(2D + 1,2)denote the halved graph of the(2D + 1)-cube.It is known that,1/2H(2D + 1,2)has two Q-polynomial structures:E0,E1,...,ED and E0,E2,E4...,E3,E1.We denote the graph 1/2H(2D +1,2)with the second Q-polynomial structure by 1/2H"(2D + 1,2).Let D denote a positive integer and N is a set with cardinality of 4D+2.The halved folded hypercube 1/2H(4D + 2,2)is defined as follows.The vertex set X is{(S,S')| S and S' is a partition of N,|S| and |S'|… are even}and two vertices(P,P'),(Q,Q')of X are adjacent whenever min{|P?Q|,|P?Q'|} = 2,where P?Q:=P?Q-P?QIn this thesis,by using the theories of Leonard pairs and universal enveloping algebra U(sl2),we determine the structures of the Terwilliger algebras of the halved hypercube 1/2H"(2D+ 1,2)and the halved folded hypercube 1/2H(4D + 2,2),respectively.