Davidson (1981) developed a general procedure, based on group representation theory, for determining the spectra of graphs distinguished by a certain rotational symmetry, with application to molecular graphs. In this paper a more general method, applicable to any arbitrarily arc weighted directed graph that has a non-trivial automorphism, and yielding both eigenvalues and eigenvectors, is developed. The proofs, elementary and straightforward, avoid the use of the theory of group characters altogether.