Testing graph isomorphism with powers of the adjacency matrix

Slightly more generally than Dan's example, if a graph is $r$-regular, the column-sum multiset of $A^s$ is $(r^s,\ldots,r^s)$. So any two nonisomorphic $r$-regular graphs on $n$ vertices provide an example.

If $A = C_3 \cup C_3$ and $B = C_6$, then it seems like the column-sum multiset is always $\{2^s,2^s,2^s,2^s,2^s,2^s\}$. I'm still interested in references and broader explanations.