On the (Non-)Existence of Tight Distance-Regular Graphs: a Local Approach

Let denote a distance-regular graph with diameter D 3. Jurišić and Vidali conjectured that if is tight with classical parameters (D, b, , ), b 2, then 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, 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 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 is not locally the block graph of an orthogonal array or a Steiner system, then the valency (and hence diameter) of is bounded by a function in the parameter b=b₁/ (1+₁), where b₁ is the intersection number of and ₁ is the second largest eigenvalue of.