On the (non-)existence of tight distance-regular graphs: a local approach

Let Γ denote a distance-regular graph with diameter D≥ 3. Jurišić and Vidali conjectured that if Γ is tight with classical parameters (D,b,α,β), b≥ 2, then Γ is not locally the block graph of an orthogonal array nor the block graph of a Steiner system. In the present paper, we prove this conjecture and, furthermore, extend it from the following aspect. Assume that for every triple of vertices x, y, z of Γ, where x and y are adjacent, and z is at distance 2 from both x and y, the number of common neighbors of x, y, z is constant. We then show that if Γ is locally the block graph of an orthogonal array (resp. a Steiner system) with smallest eigenvalue -m, m≥ 3, then the intersection number c_2 is not equal to m^2 (resp. m(m+1)). Using this result, we prove that if a tight distance-regular graph Γ is not locally the block graph of an orthogonal array or a Steiner system, then the valency (and hence diameter) of Γ is bounded by a function in the parameter b=b_1/(1+θ_1), where b_1 is the intersection number of Γ and θ_1 is the second largest eigenvalue of Γ.