A graph is called vertex and edge transitive if the automorphism group is transitive on both vertices and edges.

The spectrum of a graph is the collection (with multiplicities) of eigenvalues of the incidence matrix.

Supposedly, it is conjectured that almost all graphs have the property that they are the unique graph with their spectrum (at least, according to MathWorld).

If $\Gamma_1,\Gamma_2$ are two vertex and edge transitive graphs, with the same valence, which are isospectral (have the same spectrum) then does it follow that $\Gamma_1\cong \Gamma_2$?