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Is there examples of non-isomorphic cospectral graphs having

  1. Non-isomorphic automorphism groups?

  2. Isomorphic automorphism groups?

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Yes, there are lots of such examples of strongly regular graphs. E.g. Chang graphs give an example of for 1).

For 2), many cospectral graphs have trivial automorphism groups (thus isomorphic as groups).

Or, for a concrete example 2), take point graphs of generalised quadrangles GQ(3,3). These are two non-isomorphic cospectral strongly regular graphs on 40 vertices.

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