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 $(x-22){(x-2)}^{42}{(x+6)}^{15}{(x+8)}^2$.

中文翻译：

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18 维实等角线和互补子图的雅可比恒等式

我们证明了等角线系统的最大基数${\mathbb{右}}^{18}$最多为 59。我们的证明包括雅可比恒等式在互补子图上的新颖应用。特别地，我们证明不存在邻接矩阵具有特征多项式的图$（X-22）{（X-2）}^{42}{（X+6）}^{15}{（X+8）}^2$。