Show that a connected graph is distance-regular if and only if for each positive integer $k$, the number of $i,j$ walks of length $k$ depends only on $d(i, j)$.
I think I must use this relation $AA_i=b_{i-1}A_{i-1}+a_i A_i+c_{i+1}A_{i+1}$,when $G$ is distance-regular iff we have this relation.
if I put $i=k-1$ do I have walk of length $k$?if it is right,because $AA_{k-1}$ is in adjacency Algebra ,it is equal of linear combination of $A_{i}$ so it is only depend on d(i,j).
it will be great if you guide me.thanks a lot.