Isomorphism testing for embeddable graphs through definability
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
Weisfeiler-Lehman Refinement Requires at Least a Linear Number of Iterations
ICALP '01 Proceedings of the 28th International Colloquium on Automata, Languages and Programming,
Equivalence in finite-variable logics is complete for polynomial time
FOCS '96 Proceedings of the 37th Annual Symposium on Foundations of Computer Science
Journal of Combinatorial Theory Series B
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The k-th power of an 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(logn). 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=@W(n). In particular, this shows that for each k, there are pairs of non-isomorphic n-vertex graphs with cospectral k-th (symmetric) powers.