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!
