On strongly closed subgraphs with diameter two and the -polynomial property

In this paper, we study a distance-regular graph Γ = ( X , R ) with an intersection number a 2 ≠ 0 having a strongly closed subgraph Y of diameter 2. Let E 0 , E 1 , … , E D be the primitive idempotents corresponding to the eigenvalues θ 0 > θ 1 > ⋯ > θ D of Γ . Let V = C X be the vector space consisting of column vectors whose rows are labeled with the vertex set X . Let W be the subspace of V consisting of vectors whose supports lie in Y . A nonzero vector v ∈ W is said to be tight whenever E 0 v and at least one of E 1 v , … , E D v is zero. We show that the existence of a tight vector in W is equivalent to a balanced condition defined by P. Terwilliger. As an application, we study the structure of parallelogram-free distance-regular graphs and conditions for these graphs to be Q -polynomial.