Finite two-distance tight frames

A finite collection of unit vectors S ⊂ ℝⁿ is called a spherical two-distance set if there are two numbers a and b such that the inner products of distinct vectors from S are either a or b. We prove that if a ≠ -b, then a two-distance set that forms a tight frame for ℝⁿ is a spherical embedding of a strongly regular graph. We also describe all two-distance tight frames obtained from a given graph. Together with an earlier work by S. Waldron (2009) [22] on the equiangular case, this completely characterizes two-distance tight frames. As an intermediate result, we obtain a classification of all two-distance 2-designs.