A new proof of the Lemmens-Seidel conjecture
Chuanyuan Ge, Shiping Liu
TL;DR
The paper addresses the problem of determining the maximal number $N_{\alpha}(d)$ of equiangular lines in $\mathbb{R}^d$ with a fixed angle, focusing on the $\alpha=1/5$ case of the Lemmens-Seidel conjecture. It introduces a new proof strategy that leverages eigenvalue-multiplicity bounds for graphs with bounded degree ($\Delta_G \le 14$) and employs a matrix-projection inequality to bound the graph degree, avoiding the finite-minimal-graphs route used previously. The authors obtain a bound that matches known constructions for large $d$ (specifically showing $N_{1/5}(d) \le \left\lfloor\frac{3d-3}{2}\right\rfloor$ for $d \ge 185$), and extend the method to give a new proof for the classical $\alpha=1/3$ case. This approach broadens the toolkit for equiangular-line problems by linking Gram-matrix spectral data to graph-theoretic multiplicity bounds and could inform future work on $N_{1/(2k-1)}(d)$ for $k\ge 4$.
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 $\arccos(1/3)$.