Q: Is there a graph matrix-representation (not necessarily an $n \times n$ matrix for an $n$-graph) such that isospectrality implies graph-isomorphism? For instance, would the simple distance-matrix do the job?

Background: The 'can you hear the shape of a drum?' question can be answered of undirected graphs in the negative, but I do not know if the results are only for the adjacency and/or Laplacian matrix representations.