Sets of Complex Unit Vectors with Two Angles and Distance-Regular Graphs

TL;DR

This paper explores special sets of complex unit vectors with two fixed angles, using distance-regular graphs to construct and analyze these sets, establishing bounds and characterizations for their sizes.

Contribution

It introduces new constructions of {0,α}-sets via distance-regular graphs and characterizes graphs that achieve size bounds.

Findings

Bounds for sizes of {0,α}-sets of flat vectors

Characterization of graphs meeting size bounds

New constructions of {0,α}-sets using distance-regular graphs

Abstract

We study {0,\alpha}-sets, which are sets of unit vectors of $\mathbb{C}^m$ in which any two distinct vectors have angle 0 or \alpha. We investigate some distance-regular graphs that provide new constructions of {0,\alpha}-sets using a method by Godsil and Roy. We prove bounds for the sizes of {0,\alpha}-sets of flat vectors, and characterize all the distance-regular graphs that yield {0,\alpha}-sets meeting the bounds at equality.