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

• Published in: The Electronic journal of combinatorics Vol. 31; no. 2
• Main Authors: Koolen, Jack, Lee, Jae-Ho, Li, Shuang-Dong, Li, Yun-Han, Liang, Xiaoye, Tan, Ying-Ying
• Format: Journal Article
• Language: English
• Published: 03.05.2024
• ISSN: 1077-8926; 1077-8926
• DOI: 10.37236/12699

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 $\Gamma$.