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

Abstract

Let $Γ$ denote a distance-regular graph with diameter $D\geq 3$. Jurišić and Vidali conjectured that if $Γ$ is tight with classical parameters $(D,b,α,β)$, $b\geq 2$, 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 $-m$, $m\geq 3$, 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 $Γ$ 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 $Γ$.