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

Let \(\Gamma\) denote a distance-regular graph with diameter \(D\geq 3\). Jurišić and Vidali conjectured that if \(\Gamma\) is tight with classical parameters \((D,b,\alpha,\beta)\), \(b\geq 2\), then \(\Gamma\) 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 \(\Gamma\), 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 \(\Gamma\) is locally the block graph of an orthogonal array (resp. a Steiner system) with smallest eigenvalue \(-m\), \(m\geq 3\), 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 \(\Gamma\) is not locally the block graph of an orthogonal array or a Steiner system, then the valency (and hence diameter) of \(\Gamma\) is bounded by a function in the parameter \(b=b_1/(1+\theta_1)\), where \(b_1\) is the intersection number of \(\Gamma\) and \(\theta_1\) is the second largest eigenvalue of \(\Gamma\).