# Can distance-regular graphs with different intersection arrays have the same number of k-hop neighbors for all k?

I would like to ask a very fundamental question regarding distance-regular graphs. Denote $$d(u,v)$$ as the distance between node $$u$$ and $$v$$. Distance-regular graphs are graphs such that for any pair of vertices $$u,v$$ with $$d(u,v)=i$$, there are always $$b_i$$ neighbors $$w$$ of $$u$$ with $$d(w,v)=i+1$$ and $$c_i$$ neighbors $$w$$ of $$u$$ with $$d(w,v)=i-1$$. Therefore, distance-regular graphs are characterized by the intersection array $$(b_0,\cdots,b_{D-1};c_1,\cdots,c_D)$$ where $$D$$ is the dimension of the graph.

However, I wonder if it can be described using only the valency $$k_i$$, which is the number of nodes $$v$$ with $$d(u,v)=i$$ given an arbitrary node $$u$$. Clearly, the intersection array $$(b_0,\cdots,b_{D-1};c_1,\cdots,c_D)$$ uniquely determines $$(k_1,\cdots,k_D)$$. I would like to ask whether the converse also holds: $$(k_1,\cdots,k_D)$$ unique determines the intersection array $$(b_0,\cdots,b_{D-1};c_1,\cdots,c_D)$$. Thank you!