A graph is called half-transitive 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 half-transitive two vertex and edge transitive graphsdetermined by spectra? If not, is there some subclass which iswith the same valence, or which is known to be?are isospectral (have the same spectrum) then does it follow that $\Gamma_1\cong \Gamma_2$?