The spectral properties of signed directed graphs, which may be naturally obtained by assigning a sign to each edge of a directed graph, have received substantially less attention than those of their undirected and/or unsigned counterparts. To represent such signed directed graphs, we use a striking equivalence to ${\mathbb{T}}_6$-gain graphs to formulate a Hermitian adjacency matrix, whose entries are the unit Eisenstein integers $\exp (k\pi i/3)$, $k\in {\mathbb{Z}}_6$. Many well-known results, such as (gain) switching and eigenvalue interlacing, naturally carry over to this paradigm. We show that non-empty signed directed graphs whose spectra occur uniquely, up to isomorphism, do not exist, but we provide several infinite families whose spectra occur uniquely up to switching equivalence. Intermediate results include a classification of all signed digraphs with rank $2,3$, and a deep discussion of signed digraphs with extremely few (1 or 2) non-negative (eq. non-positive) eigenvalues.

有符号有向图的频谱特性可以通过为有向图的每条边分配一个符号而自然获得，与它们的无向和/或无符号对应物相比，受到的关注要少得多。为了表示这样的有符号有向图，我们使用了一个惊人的等价${\mathbb{吨}}_6$- 增益图以制定 Hermitian 邻接矩阵，其条目是单位 Eisenstein 整数 $\mathrm{经验值}(克\pi \mathrm{一世}/3)$, $克\in {\mathbb{Z}}_6$. 许多众所周知的结果，例如（增益）切换和特征值交错，自然会延续到这种范式。我们证明了非空有符号有向图的光谱唯一出现，直到同构，不存在，但我们提供了几个无限族，它们的光谱在切换等价之前唯一出现。中间结果包括所有带秩的有符号有向图的分类$2,3$，以及对具有极少（1 或 2）非负（即非正）特征值的有符号有向图的深入讨论。