A new proof of the Lemmens-Seidel conjecture

TL;DR

This paper presents a novel proof of the Lemmens-Seidel conjecture on the maximum number of equiangular lines with a specific angle, using eigenvalue multiplicity bounds and matrix projection techniques, offering new insights into classical and modern results.

Contribution

The paper introduces a new proof of the Lemmens-Seidel conjecture using eigenvalue bounds and matrix techniques, providing a different approach from previous analyses.

Findings

Validated the maximum number of equiangular lines with angle arccos(1/5).

Provided a new proof for the classical case with angle arccos(1/3).

Applied eigenvalue multiplicity bounds to graph degree constraints.

Abstract

In this paper, we give a new proof of the Lemmens-Seidel conjecture on the maximum number of equiangular lines with a common angle $\arccos(1/5)$. This conjecture was previously resolved by Cao, Koolen, Lin, and Yu in 2022 through an analysis involving forbidden subgraphs for the smallest Seidel eigenvalue $-5$. Our new proof is based on bounds on eigenvalue multiplicities of graphs with degree no larger than $14$. To control the maximum degree of the graph associated with equiangular lines, we employ a recent inequality of Balla derived by matrix projection techniques. Our strategy also leads to a new proof for the classical result obtained by Lemmens and Seidel in 1973 for the case where the common angle is $\arccos(1/3)$.