How to use spectral graph theory to get a measure for graph symmetry? I looked at graphs, like $K_{12}$ or Frucht's graph and wondered if their spectrum, more specific the degenercies of their eigenvalues, is a mesaure for the (a)symmetry of the corresponding graph?
For the examples given it works out quite nice:
$$
\begin{eqnarray}
\text{Graph} && \text{Spectrum (approximated)}\\
\hline
K_{12} && (-1)^{11}11^1\\
\text{Frucht} && (-2.33)^1 (-2)^1 (-1.80)^1 (-1.45)^1 (-1)^1 (-0.44)^1 \\
&&0^1\\&& 0.51^1 1.24^1 2^1 2.27^1 3^1
\end{eqnarray}
$$
The asymmetric cubic graph of Frucht is an identity graph, i.e. the graph automorphism group contains only the identity, and also shows lowest symmetry in its eigenvalues.

Basically, having a large automorphism group tends to push the multiplicities up.
There is an old result that the eigenvalues of a vertex-transitive graph cannot all be simple. Each eigenspace of the graph provides a representation of the automorphism group, and so if the group has no faithful representation of low degree the multiplicities go up. 
To view it another way, the permutation matrices in the automorphism group lie in the algebra of matrices that commute with the adjacency matrix $A$, and the dimension of this is the sum of the squares of its eigenvalue multiplicities.
As Randy E notes the converse is not true, we can have large multiplicities and trivial group.
