Equiangular tight frames with centroidal symmetry

TL;DR

This paper explores centroidal symmetry in equiangular tight frames (ETFs), linking them to strongly regular graphs, and provides new existence and non-existence results for certain SRGs.

Contribution

It introduces centroidal symmetry in ETFs, connects them to SRGs, and proves the existence or non-existence of specific SRGs based on this relationship.

Findings

Established a new equivalence between centroid-symmetric ETFs and SRGs.

Proved the existence of certain SRGs using ETF properties.

Disproved the existence of other SRGs based on the symmetry connection.

Abstract

An equiangular tight frame (ETF) is a set of unit vectors whose coherence achieves the Welch bound, and so is as incoherent as possible. Though they arise in many applications, only a few methods for constructing them are known. Motivated by the connection between real ETFs and graph theory, we introduce the notion of ETFs that are symmetric about their centroid. We then discuss how well-known constructions, such as harmonic ETFs and Steiner ETFs, can have centroidal symmetry. Finally, we establish a new equivalence between centroid-symmetric real ETFs and certain types of strongly regular graphs (SRGs). Together, these results give the first proof of the existence of certain SRGs, as well as the disproofs of the existence of others.