  The k-th power of a n-vertex graph X is the iterated cartesian product of X
with itself. The k-th symmetric power of X is the quotient graph of certain
subgraph of its k-th power by the natural action of the symmetric group. It is
natural to ask if the spectrum of the k-th power --or the spectrum of the k-th
symmetric power-- is a complete graph invariant for small values of k, for
example, for k=O(1) or k=O(log n). In this paper, we answer this question in
the negative: we prove that if the well known 2k-dimensional Weisfeiler-Lehman
method fails to distinguish two given graphs, then their k-th powers --and
their k-th symmetric powers-- are cospectral. As it is well known, there are
pairs of non-isomorphic n-vertex graphs which are not distinguished by the
k-dim WL method, even for k=Omega(n). In particular, this shows that for each
k, there are pairs of non-isomorphic n-vertex graphs with cospectral k-th
(symmetric) powers.
