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

In this paper, we study a distance-regular graph Γ=(X,R)Γ=(X,R) with an intersection number a2≠0a2≠0 having a strongly closed subgraph YY of diameter 2. Let E0,E1,…,EDE0,E1,…,ED be the primitive idempotents corresponding to the eigenvalues θ0>θ1>⋯>θDθ0>θ1>⋯>θD of ΓΓ. Let V=CX be the vector space consisting of column vectors whose rows are labeled with the vertex set XX. Let WW be the subspace of VV consisting of vectors whose supports lie in YY. A nonzero vector v∈W is said to be tight whenever E0v and at least one of E1v,…,EDv is zero. We show that the existence of a tight vector in WW 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 QQ-polynomial.