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We derive easily verifiable conditions which characterize when complex Seidelmatrices containing cube roots of unity have exactly two eigenvalues. Theexistence of such matrices is equivalent to the existence of equiangular tightframes for which the inner product between any two frame vectors is always acommon multiple of the cube roots of unity. We also exhibit a relationshipbetween these equiangular tight frames, complex Seidel matrices, and highlyregular, directed graphs. We construct examples of such frames with arbitrarilymany vectors.

Equiangular tight frames from complex Seidel matrices containing cube roots of unity