Real equiangular lines in dimension 18 and the Jacobi identity for complementary subgraphs

TL;DR

This paper establishes an upper bound of 59 for the maximum number of equiangular lines in 18-dimensional space using a novel approach involving the Jacobi identity for complementary subgraphs.

Contribution

It introduces a new application of the Jacobi identity to the problem of equiangular lines, providing a key nonexistence result for certain graph spectra.

Findings

Maximum equiangular lines in R^18 is at most 59.

Proves nonexistence of a graph with specific characteristic polynomial.

Introduces a novel method using the Jacobi identity for complementary subgraphs.

Abstract

We show that the maximum cardinality of an equiangular line system in $\mathbb R^{18}$ is at most $59$. Our proof includes a novel application of the Jacobi identity for complementary subgraphs. In particular, we show that there does not exist a graph whose adjacency matrix has characteristic polynomial $(x-22)(x-2)^{42} (x+6)^{15} (x+8)^2$.