Taut distance-regular graphs and the subconstituent algebra Original Research Article

We consider a bipartite distance-regular graph image with diameter image and valency image. Let X denote the vertex set of image and fix image. Let image denote the graph with vertex set image, and edge set image, where image is the path-length distance function for image. The graph image has exactly image vertices, where image is the second valency of image. Let image denote the eigenvalues of the adjacency matrix of image; we call these the local eigenvalues of image. Let A denote the adjacency matrix of image. We obtain upper and lower bounds for the local eigenvalues in terms of the intersection numbers of image and the eigenvalues of A. Let image denote the subalgebra of image generated by image, where for image, image represents the projection onto the imageth subconstituent of image with respect to x. We refer to T as the subconstituent algebra (or Terwilliger algebra) of image with respect to x. An irreducible T-module W is said to be thin whenever image for image. By the endpoint of W we mean image. We give a detailed description of the thin irreducible T-modules that have endpoint 2 and dimension image. MacLean [An inequality involving two eigenvalues of a bipartite distance-regular graph, Discrete Math. 225 (2000) 193–216] defined what it means for image to be taut. We obtain three characterizations of the taut condition, each of which involves the local eigenvalues or the above T-modules.