TITLE: Are vertex and edge-transitive graphs determined by their spectrum?
QUESTION [8 upvotes]: 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 $\Gamma_1\cong \Gamma_2$?

REPLY [4 votes]: As Chris said, the answer is most probably no, and most likely there are known such examples. There are many examples and constructions of non isomorphic graphs with the same spectrum and some of these examples are Cayley graphs and other very symmetric graphs. Cayley graphs are always vertex transitive and quite often, for a suitable choise of generators, also edge transitive. Some of these examples are based on a famous paper 
of Sunada. Sunada's method was originally for creating isospectral manifolds but it can be applied (and is even easier) to create isospectral graphs. The following paper by Bob Brooks "Isospecrtal graphs and isospectral surfaces" is a good starting point.