Maximality of Seidel matrices and switching roots of graphs

TL;DR

This paper classifies maximal Seidel matrices with eigenvalue 3, linking them to maximal equiangular lines, and introduces the concept of strong maximality inspired by the E8 root system.

Contribution

It provides a classification of maximal Seidel matrices with eigenvalue 3 and introduces the notion of strong maximality related to the absolute bound.

Findings

Classified all maximal Seidel matrices with eigenvalue 3.

Connected maximal Seidel matrices to maximal equiangular lines.

Defined and characterized strongly maximal Seidel matrices.

Abstract

In this paper, we discuss maximality of Seidel matrices with a fixed largest eigenvalue. We present a classification of maximal Seidel matrices of largest eigenvalue $3$, which gives a classification of maximal equiangular lines in a Euclidean space with angle $\arccos1/3$. Motivated by the maximality of the exceptional root system $E_8$, we define strong maximality of a Seidel matrix, and show that every Seidel matrix achieving the absolute bound is strongly maximal.