Sets of complex unit vectors with few inner products and distance-regular graphs

We study { 0 , α } -sets, which are sets of unit vectors of C m in which the inner product of any two distinct vectors has absolute value 0 or α . We investigate some distance-regular graphs that provide new constructions of { 0 , α } -sets using a method by Godsil and Roy. We prove bounds for the sizes of { 0 , α } -sets of flat vectors, and characterize all the distance-regular graphs that yield { 0 , α } -sets meeting the bounds at equality.