Complex Two-Graphs via Equiangular Tight Frames

TL;DR

This paper generalizes the concept of two-graphs by incorporating roots of unity beyond
, establishing new connections with equiangular tight frames and extending classical graph theory relationships.

Contribution

It introduces a novel generalization of two-graphs using roots of unity, expanding the framework for equiangular tight frames and related combinatorial structures.

Findings

Established a correspondence between generalized two-graphs and equiangular tight frames.

Extended classical connections between graphs, two-graphs, and equiangular lines.

Provided a new mathematical framework for analyzing complex two-graphs.

Abstract

In `A survey of two-graphs' \cite{Sei}, J.J. Seidel lays out the connections between simple graphs, two-graphs, equiangular lines and strongly regular graph. It is well known that there is a one-to-one correspondence between regular two-graphs and equiangular tight frames. This article gives a generalization of two-graphs for which these connections can be mimicked using roots of unity beyond $\pm 1$.