Spectra of symmetric powers of graphs and the Weisfeiler-Lehman refinements

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 = Ω(n). In particular, this shows that for each k, there are pairs of non-isomorphic n-vertex graphs with cospectral k-th (symmetric) powers.