Non-Isomorphic Graphs with Cospectral Symmetric Powers

The symmetric m-th power of a graph is the graph whose vertices are m-subsets of vertices and in which two m-subsets are adjacent if and only if their symmetric difference is an edge of the original graph. It was conjectured that there exists a fixed m such that any two graphs are isomorphic if and only if their m-th symmetric powers are cospectral. In this paper we show that given a positive integer m there exist infinitely many pairs of non-isomorphic graphs with cospectral m-th symmetric powers. Our construction is based on theory of multidimensional extensions of coherent configurations.