半超立方体和半折叠超立方体上的Terwilliger代数结构

发布时间：2019-02-28 08:47

【摘要】：设r=(X,R)是一个直径大于等于3的有限连通二部图.定义图Γ2如下:其顶点集合为X,两个顶点x,y相邻当且仅当在r中(?)(x,y)=2.易知,图Γ2有两个连通分支.Γ2在每一个连通分支上诱导出的子图叫做r的半图,记为1/2Γ.将超立方体H(2D+1,2)的半图记为1/2H(2D+1,2),熟知,1/2H(2D+1,2)有两个Q-多项式结构,分别为E。,E1,...,ED和E0,E2,E4...,E3,E1.把具有第二个Q-多项式结构的图1/2H(2D+1,2)记为1/2H"(2D+1,2).设D是一个正整数,N是一个基数为4D +2的集合.半折叠超立方体1/2H(4D +2,2)定义如下:其顶点集合为X={(S,S')|S和S'是N的一个分拆,S和S'的基数均为偶数}.X中的两个顶点(P,P'),(Q,Q')相邻当且仅当min{|P△Q|,|P△Q'|= 2,其中P△Q:=P∪Q-P∩Q.本文利用Leonard对,泛包络代数U(sl2)等理论分别刻画了半超立方体1/2H"(2D+1,2)和半折叠超立方体1/2H(4D + 2,2)的Terwilliger代数结构。
[Abstract]:Let r = (X, R) be a finite connected bipartite graph with a diameter greater than or equal to 3. Define graph 螕 2 as follows: its vertex set is X, two vertices x, y adjacent if and only if in r (?) (x, y) = 2. It is easy to know that a graph 螕 2 has two connected branches. The subgraph induced by 螕 2 on each connected component is called a semi-graph of r, denoted as 1 ~ 2 螕. The semi-graph of hypercube H (2D 1, 2) is denoted as 1? 2H (2D 1, 2). As is known, 1? 2H (2D 1, 2) has two Q-polynomial structures: E., E1,. Ed and E0, E2, E4, E3, E1. The graph 1 / 2 H (2D 1, 2) with the structure of the second Q-polynomial is denoted as 1 * 2H "(2D 1, 2). Let D be a positive integer and N be a set of cardinality 4D 2. A semi-folded hypercube 1 / 2H (4D 2, 2) is defined as follows: its vertex set is X = {(S, S') | S and S 'are a split of N, and S' are cardinality of two vertices in even} .X (P, P'), (Q,). Q') adjacent if and only if min {| P Q |, | P Q'| = 2, where P Q = 2 In this paper, we use the theory of Leonard pair and universal envelope algebra U (sl2) to characterize the Terwilliger algebraic structures of semi-hypercube 1 / 2 H "(2D 1, 2) and semi-folded hypercube 1 / 2 H (4D 2, 2), respectively.
【学位授予单位】：河北师范大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：O157

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