Equiangular Lines and Covers of the Complete Graph

TL;DR

This paper explores the connection between complex equiangular lines meeting the Welch bound and distance-regular covers of the complete graph, revealing new constructions and bounds for such configurations.

Contribution

It extends the known relationship to complex lines with Gram matrices having prime roots of unity and derives bounds for abelian distance-regular covers.

Findings

Construction of antipodal distance-regular graphs from equiangular lines meeting the Welch bound.

Use of the Gerzon bound to establish parameter bounds for abelian covers.

Demonstration of how prime roots of unity in Gram matrices relate to graph structures.

Abstract

The relation between equiangular sets of lines in the real space and distance-regular double covers of the complete graph is well known and studied since the work of Seidel and others in the 70's. The main topic of this paper is to continue the study on how complex equiangular lines relate to distance-regular covers of the complete graph with larger index. Given a set of equiangular lines meeting the relative (or Welch) bound, we show that if the entries of the corresponding Gram matrix are prime roots of unity, then these lines can be used to construct an antipodal distance-regular graph of diameter three. We also study in detail how the absolute (or Gerzon) bound for a set of equiangular lines can be used to derive bounds of the parameters of abelian distance-regular covers of the complete graph.