Spherical two-distance sets

A set S of unit vectors in n-dimensional Euclidean space is called spherical two-distance set, if there are two numbers a and b, and inner products of distinct vectors of S are either a or b. The largest cardinality g(n) of spherical two-distance sets is not exceed n(n+3)/2. This upper bound is known to be tight for n=2,6,22. The set of mid-points of the edges of a regular simplex gives the lower bound L(n)=n(n+1)/2 for g(n. In this paper using the so-called polynomial method it is proved that for nonnegative a+b the largest cardinality of S is not greater than L(n). For the case a+b<0 we propose upper bounds on |S| which are based on Delsarte's method. Using this we show that g(n)=L(n) for 6<n<22, 23<n<40, and g(23)=276 or 277.Comment: 9 pages, (v2) several small changes and corrections suggested by referees, accepted in Journal of Combinatorial Theory, Series

Spectral conditions for spherical two-distance sets

A set of points $S$ in $d$-dimensional Euclidean space $\mathbb{R}^d$ is called a 2-distance set if the set of pairwise distances between the points has cardinality two. The 2-distance set is called spherical if its points lie on the unit sphere in $\mathbb{R}^{d}$. We characterize the spherical 2-distance sets using the spectrum of the adjacency matrix of an associated graph and the spectrum of the projection of the adjacency matrix onto the orthogonal complement of the all-ones vector. We also determine the lowest dimensional space in which a given spherical 2-distance set could be represented using the graph spectrum.Comment: 12 pages, 2 table

Spherical two-distance sets and eigenvalues of signed graphs

We study the problem of determining the maximum size of a spherical two-distance set with two fixed angles (one acute and one obtuse) in high dimensions. Let $N_{\alpha,\beta}(d)$ denote the maximum number of unit vectors in $\mathbb R^d$ where all pairwise inner products lie in $\{\alpha,\beta\}$. For fixed $-1\leq\beta<0\leq\alpha<1$, we propose a conjecture for the limit of $N_{\alpha,\beta}(d)/d$ as $d \to \infty$ in terms of eigenvalue multiplicities of signed graphs. We determine this limit when $\alpha+2\beta<0$ or $(1-\alpha)/(\alpha-\beta) \in \{1, \sqrt{2}, \sqrt{3}\}$. Our work builds on our recent resolution of the problem in the case of $\alpha = -\beta$ (corresponding to equiangular lines). It is the first determination of $\lim_{d \to \infty} N_{\alpha,\beta}(d)/d$ for any nontrivial fixed values of $\alpha$ and $\beta$ outside of the equiangular lines setting.Comment: 21 pages, 8 figure

Spherical two-distance sets and related topics in harmonic analysis

This dissertation is devoted to the study of applications of harmonic analysis. The maximum size of spherical few-distance sets had been studied by Delsarte at al. in the 1970s. In particular, the maximum size of spherical two-distance sets in $\mathbb{R}^n$ had been known for $n \leq 39$ except $n=23$ by linear programming methods in 2008. Our contribution is to extend the known results of the maximum size of spherical two-distance sets in $\mathbb{R}^n$ when $n=23$, $40 \leq n \leq 93$ and $n \neq 46, 78$. The maximum size of equiangular lines in $\mathbb{R}^n$ had been known for all $n \leq 23$ except $n=14, 16, 17, 18, 19$ and $20$ since 1973. We use the semidefinite programming method to find the maximum size for equiangular line sets in $\mathbb{R}^n$ when $24 \leq n \leq 41$ and $n=43$. We suggest a method of constructing spherical two-distance sets that also form tight frames. We derive new structural properties of the Gram matrix of a two-distance set that also forms a tight frame for $\mathbb{R}^n$. One of the main results in this part is a new correspondence between two-distance tight frames and certain strongly regular graphs. This allows us to use spectral properties of strongly regular graphs to construct two-distance tight frames. Several new examples are obtained using this characterization. Bannai, Okuda, and Tagami proved that a tight spherical designs of harmonic index 4 exists if and only if there exists an equiangular line set with the angle $\arccos (1/(2k-1))$ in the Euclidean space of dimension $3(2k-1)^2-4$ for each integer $k \geq 2$. We show nonexistence of tight spherical designs of harmonic index $4$ on $S^{n-1}$ with $n\geq 3$ by a modification of the semidefinite programming method. We also derive new relative bounds for equiangular line sets. These new relative bounds are usually tighter than previous relative bounds by Lemmens and Seidel