Taut distance-regular graphs and the subconstituent algebra

Abstract

We consider a bipartite distance-regular graph Γ with diameter D ⩾ 4 and valency k ⩾ 3 . Let X denote the vertex set of Γ and fix x ∈ X . Let Γ 2 2 denote the graph with vertex set X ˘ = { y ∈ X | ∂ ( x , y ) = 2 } , and edge set R ˘ = { yz | y , z ∈ X ˘ , ∂ ( y , z ) = 2 } , where ∂ is the path-length distance function for Γ . The graph Γ 2 2 has exactly k 2 vertices, where k 2 is the second valency of Γ . Let η 1 , η 2 , … , η k 2 denote the eigenvalues of the adjacency matrix of Γ 2 2 ; we call these the local eigenvalues of Γ . Let A denote the adjacency matrix of Γ . We obtain upper and lower bounds for the local eigenvalues in terms of the intersection numbers of Γ and the eigenvalues of A. Let T = T ( x ) denote the subalgebra of Mat X ( C ) generated by A , E 0 * , E 1 * , … , E D * , where for 0 ⩽ i ⩽ D , E i * represents the projection onto the i th subconstituent of Γ with respect to x. We refer to T as the subconstituent algebra (or Terwilliger algebra) of Γ with respect to x. An irreducible T-module W is said to be thin whenever dim E i * W ⩽ 1 for 0 ⩽ i ⩽ D . By the endpoint of W we mean min { i | E i * W ≠ 0 } . We give a detailed description of the thin irreducible T-modules that have endpoint 2 and dimension D - 3 . MacLean [An inequality involving two eigenvalues of a bipartite distance-regular graph, Discrete Math. 225 (2000) 193–216] defined what it means for Γ to be taut. We obtain three characterizations of the taut condition, each of which involves the local eigenvalues or the above T-modules.