We characterize maximal digraphs whose Hermitian adjacency matrix has spectral radius at most 2. This can be regarded as a generalization of the classical result on undirected graphs whose adjacency matrix has spectral radius 2. This classical result is due to Smith, Lemmens and Seidel, while Hermitian adjacency matrices were introduced by Liu and Li, Guo and Mohar independently. Unlike the undirected case where the results are just (extended) Dynkin diagrams of type A, D, or E, we have more complicated digraphs, some infinite families and finitely many sporadic ones. The characterization is obtained by translating the classification of cyclotomic matrices due to Greaves into that of Hermitian adjacency matrices. Passing to the lattice over the ring of Gaussian integers with Gram matrix $H+2I$, where $H$ is the Hermitian adjacency matrix, and then restricting the coefficients to the integers, we will find correspondence to the classical result in terms of root lattices.

This is joint work with Alexander Gavrilyuk.