TITLE: Generalised Isospectrality of Graphs
QUESTION [6 upvotes]: 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.

REPLY [2 votes]: Distance matrix is not going to help, as examples of non-isomorphic co-spectral strongly regular graphs would tell you.
The smallest examples like this exist on 16 vertices: Shrikhande graph.
In general, there seem to be logic-related obstacles for non-isomorphism certificates of this sort, related to 1st order logics with counting. This has started with the famous paper 
An Optimal Lower Bound on the Number of
Variables for Graph Identification by Cai, Furer and Immerman.