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 str...

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Bibliographic Details
Published in:Linear algebra and its applications 2015-06, Vol.475, p.163-175
Main Authors: Barg, Alexander, Glazyrin, Alexey, Okoudjou, Kasso A., Yu, Wei-Hsuan
Format: Article
Language:English
Subjects:
Finite tight frames
Spherical 2-designs
Spherical designs of harmonic index 2
Spherical two-distance sets
Strongly regular graphs
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Summary: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.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2015.02.020