Bounding the intersection number c2 of a distance-regular graph with classical parameters (D,b,α,β) in terms of b

Let Γ be a distance-regular graph with classical parameters $(D,b,\alpha ,\beta )$ and $b\ge 1$. It is known that Γ is Q-polynomial with respect to ${\theta }_1$, where ${\theta }_1=\frac{b_1}{b}-1$ is the second largest eigenvalue of Γ. And it was shown that for a distance-regular graph Γ with classical parameters $(D,b,\alpha ,\beta )$, $D\ge 5$ and $b\ge 1$, if $a_1$ is large enough compared to b and Γ is thin, then the intersection number $c_2$ of Γ is bounded above by a function of b. In this paper, we obtain a similar result without the assumption that the graph Γ is thin.