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

Abstract

Let $\Gamma$ denote a distance-regular graph with diameter $D\ge 3$. Jurišić and Vidali conjectured that if $\Gamma$ is tight with classical parameters $(D,b,\alpha ,\beta )$, $b\ge 2$, 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 $-m$, $m\ge 3$, 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 $\Gamma$.

18 pages

Cited by in corpus (1)

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