2014/02/14 by Alexander Barg, Alexei Glazyrin, Barg, Alexander +5 · 3 citations
Mathematics · Computer Science · #Mathematical Approximation and Integration #Mathematical Analysis and Transform Methods #Digital Image Processing Techniques
paper · pdf · doi:10.48550/arxiv.1402.3521
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, and every strongly regular graph gives rise to two-distance tight frames through standard spherical embeddings. Together with an earlier work by S. Waldron on the equiangular case (\em 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.\