The Terwilliger polynomial of a Q-polynomial distance-regular graph and its application to pseudo-partition graphs

Let Γ be a Q-polynomial distance-regular graph with diameter at least 3. Terwilliger (1993) implicitly showed that there exists a polynomial, say T(λ)∈R[λ], of degree 4 depending only on the intersection numbers of Γ and its Q-polynomial ordering and such that T(η)≥0 holds for any non-principal eigenvalue η of the local graph Γ(x) for any vertex x∈V(Γ).We call T(λ) the Terwilliger polynomial of Γ. In this paper, we give an explicit formula for T(λ) in terms of the intersection numbers of Γ and the dual eigenvalues of Γ with respect to the first primitive idempotent in its Q-polynomial ordering. We then apply this polynomial to show that all pseudo-partition graphs with diameter at least 3 are known.