Equiangular line systems and switching classes containing regular graphs

TL;DR

This paper investigates the properties of equiangular line systems and their associated Seidel matrices, focusing on conditions for regular graphs in switching classes and establishing an upper bound for line systems in 18-dimensional space.

Contribution

It provides new necessary and sufficient conditions for regular graphs in switching classes of Seidel matrices with three eigenvalues and establishes an upper bound for equiangular lines in 18 dimensions.

Findings

Necessary and sufficient conditions for regular graphs in switching classes.

Upper bound of 60 for equiangular lines in 18-dimensional space.

Progress towards understanding the structure of equiangular line systems.

Abstract

We develop the theory of equiangular lines in Euclidean spaces. Our focus is on the question of when a Seidel matrix having precisely three distinct eigenvalues has a regular graph in its switching class. We make some progress towards an answer to this question by finding some necessary conditions and some sufficient conditions. Furthermore, we show that the cardinality of an equiangular line system in $18$ dimensional Euclidean space is at most $60$.