The real equiangular tight frames obtained from rank $3$ graphs

TL;DR

This paper classifies all nontrivial real equiangular tight frames derived from primitive rank 3 graphs and their associated strongly regular graphs, enriching the understanding of their geometric and combinatorial structures.

Contribution

It provides a complete classification of real equiangular tight frames from primitive rank 3 graphs and related strongly regular graphs, which was previously unknown.

Findings

All such frames are explicitly characterized.

Connections between frames and primitive rank 3 graphs are established.

New examples of equiangular tight frames are identified.

Abstract

We present all nontrivial real equiangular tight frames $\{\varphi_m\}_{m=1}^M$ in $\mathbb{R}^N$ obtained as spherical embeddings of primitive rank $3$ graphs on $M$ vertices, and those such that one of their associated $M$ strongly regular graphs on $M-1$ vertices is a primitive rank $3$ graph.