论文标题
签名有向图的光谱基础和表征
Spectral Fundamentals and Characterizations of Signed Directed Graphs
论文作者
论文摘要
签名的有向图的光谱特性可以自然地通过向有向图的每个边缘分配一个符号来自然获得，但其注意力比其未指向和/或未签名的对应物的注意力要少得多。要代表此类签名的有向图，我们使用与$ \ Mathbb {t} _6 $ Gain图的醒目等效性来制定Hermitian邻接矩阵，其条目是Eisenstein Integers $ \ exp（Kπi/3），Kistem $ $ $ k \ in \ Mathbbbb {Z} ^ $ nock and-nock and feale and feale and fep）特征值交织，自然而然地转移到了这个范式上。我们表明，非空的频谱出现的非空签名的有向图，直到同构不存在，但我们提供了几个无限的家族，这些家族的光谱出现在唯一的转换等效性方面。中级结果包括对所有签名的挖掘物的分类，其等级$ 2,3 $，以及对签名的Digraphs的深入讨论，这些图形很少（1或2）非负（等式非阳性）特征值。
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 $\exp(kπi/3),$ $k\in \mathbb{Z}_6.$ 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.
