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 c₂ is not equal to m² (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=b₁/(1+θ₁), where b₁ is the intersection number of Γ and θ₁ is the second largest eigenvalue of Γ.