Graph representations, two-distance sets, and equiangular lines

New Bounds for Equiangular Lines and Spherical Two-Distance Sets

The set of points in a metric space is called an s-distance set if pairwise distances between these points admit only s distinct values. Two-distance spherical sets with the set of scalar products {α,−α}, α ∈ [0, 1), are called equiangular. The problem of determining the maximal size of s-distance sets in various spaces has a long history in mathematics. We determine a new method of bounding th...

Different-Distance Sets in a Graph

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...

Some remarks on Heisenberg frames and sets of equiangular lines

We consider the long standing problem of constructing d equiangular lines in C, i.e., finding a set of d unit vectors (φj) in C d with |〈φj , φk〉| = 1 √ d + 1 , j 6= k. Such ‘equally spaced configurations’ have appeared in various guises, e.g., as complex spherical 2–designs, equiangular tight frames, isometric embeddings `2(d) → `4(d), and most recently as SICPOVMs in quantum measurement theor...

Large Equiangular Sets of Lines in Euclidean Space

A construction is given of 29(d + 1) 2 equiangular lines in Euclidean d-space, when d = 3 · 22t−1 − 1 with t any positive integer. This compares with the well known “absolute” upper bound of 12d(d+ 1) lines in any equiangular set; it is the first known constructive lower bound of order d2 . For background and terminology we refer to Seidel [3]. The standard method for obtaining a system of equi...

Equiangular lines in Euclidean spaces