For a regular graph, eigenvalues and <a href="https://en.wikipedia.org/wiki/Laplacian_matrix">Laplacian eigenvalues</a> coincide.  I believe that in Fan Chung's book <i>Spectral Graph Theory</i> it is proved that if $d$ is the diameter of the graph, then it has at least $d+1$ distinct Laplacian eigenvalues.  (This bound is tight, and is achieved for example by <a href="https://en.wikipedia.org/wiki/Distance-regular_graph">distance-regular graphs</a>.)  So in particular, having a lot of repeated eigenvalues does not necessarily indicate a lot of automorphisms; it might just mean that the diameter is small. For example, a <a href="https://en.wikipedia.org/wiki/Strongly_regular_graph#Eigenvalues">strongly regular graph</a> has only three distinct eigenvalues but <a href="https://mathoverflow.net/q/41194">often has trivial automorphism group</a>.