FINITE TWO-DISTANCE TIGHT FRAMES

ABSTRACT. 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 6 = −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 on the equiangular case (Linear Alg. Appl., vol. 41, pp. 2228-2242, 2009) this completely characterizes two-distance tight frames. As an intermediate result, we obtain a classification of all two-distance 2-designs. 1