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

Abstract

We show that the maximum cardinality of an equiangular line system in R18 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.