On bipartite distance-regular graphs with exactly two irreducible T-modules with endpoint two

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• 作者：Mark S. MacLean^a ; ^{macleanm@seattleu.edu} ; &Scaron ; tefko Miklavič^b ; ¹ ; ^{stefko.miklavic@upr.si}
• 关键词：05E30
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：515
• 期：Complete
• 页码：275-297
• 全文大小：486 K
• 卷排序：515

文摘

Let Γ denote a bipartite distance-regular graph with diameter D≥4D≥4 and valency k≥3k≥3. Let X denote the vertex set of Γ, and let A   denote the adjacency matrix of Γ. For x∈Xx∈X let T=T(x)T=T(x) denote the subalgebra of MatX(C)MatX(C) generated by A  , E0⁎,E1⁎,…,ED⁎, where for 0≤i≤D0≤i≤D, Ei⁎ represents the projection onto the ith subconstituent of Γ with respect to x. We refer to T as the Terwilliger algebra of Γ with respect to x. An irreducible T-module W is said to be thin   whenever dim Ei⁎W≤1 for 0≤i≤D0≤i≤D. By the endpoint of W   we mean min{i|Ei⁎W≠0}. For 0≤i≤D0≤i≤D, let Γi(z)Γi(z) denote the set of vertices in X that are distance i from vertex z  . Define a parameter Δ2Δ2 in terms of the intersection numbers by Δ2=(k−2)(c3−1)−(c2−1)p222. In this paper we prove the following are equivalent: (i) Δ2>0Δ2>0 and for 2≤i≤D−22≤i≤D−2 there exist complex scalars αi,βiαi,βi with the following property: for all x,y,z∈Xx,y,z∈X such that ∂(x,y)=2∂(x,y)=2, ∂(x,z)=i∂(x,z)=i, ∂(y,z)=i∂(y,z)=i we have αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|αi+βi|Γ1(x)∩Γ1(y)∩Γi−1(z)|=|Γi−1(x)∩Γi−1(y)∩Γ1(z)|; (ii) For all x∈Xx∈X there exist up to isomorphism exactly two irreducible modules for the Terwilliger algebra T(x)T(x) with endpoint two, and these modules are thin.