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Let G be a distance-regular graph. Let A denote the adjacency matrix of G. Fix a vertex x of G. For each i(0⩽i⩽D), let Ei∗=Ei∗(x) denote the projection onto the ith subconstituent of G with respect to x. Let T(x) denote the C-algebra generated by A and {Ei∗|0⩽i⩽D}. We call T(x) the Terwilliger algebra of G with respect to x. An irreducible T(x)-module W is said to be thin if dim Ei∗W⩽1 for 0⩽i⩽D....
Let Γ denote a bipartite distance-regular graph with diameter D>=4 and valency k>=3. Let θ,θ' denote eigenvalues of Γ other than k and -k. We obtain an inequality involving θ,θ' and the intersection numbers of Γ, which we refer to as the bipartite fundamental bound (BFB). Let E,F denote the primitive idempotents of Γ associated with θ,θ', respectively. We show that the following are equivalent:...
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