On the Koolen–Park inequality and Terwilliger graphs

Abstract

J.H. Koolen and J. Park proved a lower bound for the intersection number $c_2$ of a distance-regular graph $\Gamma$. Moreover, they showed that a graph $\Gamma$, for which equality is attained in this bound, is a Terwilliger graph. We prove that $\Gamma$ is the icosahedron, the Doro graph or the Conway–Smith graph if equality is attained and $c_2\ge 2$.