On a Koolen -- Park inequality and Terwilliger graphs

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