In this paper we consider a distance-regular graph Γ. Fix a vertex x of Γ and consider the corresponding subconstituent algebra T . The algebra T is the C-algebra generated by the Bose-Mesner algebra M of Γ and the dual Bose-Mesner algebra M * of Γ with respect to x. We consider the subspaces along with their intersections and sums. In our notation, M M * means Span{RS|R ∈ M, S ∈ M * }, and so on. We introduce a diagram that describes how these subspaces are related. We describe in detail that part of the diagram up to M M * + M * M . For each subspace U shown in this part of the diagram, we display an orthogonal basis for U along with the dimension of U . For an edge U ⊆ W from this part of the diagram, we display an orthogonal basis for the orthogonal complement of U in W along with the dimension of this orthogonal complement.
We consider a primitive distance-regular graph Γ with diameter at least 3. We use the intersection numbers of Γ to find a positive semidefinite matrix G with integer entries. We show that G has determinant zero if and only if Γ is Q-polynomial.
We prove that a distance-regular graph with intersection array {22,16,5;1,2,20} does not exist. To prove this, we assume that such a graph exists and derive some combinatorial properties of its local graph. Then we construct a partial linear space from the local graph to display the contradiction.
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