AI 中文总结
该研究发展了$\boldsymbol{\top}$-增益有向图的增益邻接、拉普拉斯等矩阵的谱理论，扩展了相关经典结果，确定了单圈有向图的谱，并刻画了圈平衡等性质。
AI 中文摘要
$\boldsymbol{\top}$-增益有向图是一类在弧上带有复单位增益的有向图，对反向弧无任何限制。该框架统一了多种图类型，如符号图、混合图、复单位增益图、有向图及符号有向图。增益邻接矩阵、拉普拉斯矩阵和无符号拉普拉斯矩阵通常是非厄米矩阵，具有复谱。我们发展了这些矩阵的谱理论，扩展了符号有向图和复单位增益图的经典结果。对于增益邻接矩阵，我们建立了行列式和特征多项式公式，通过切换等价性和与基础有向图的同谱性刻画了圈平衡性。我们进一步根据基础有向图及其最大出度对谱半径进行了界定，当达到等号时由$\boldsymbol{\top}$-平衡性刻画。作为结果，我们确定了$\boldsymbol{\top}$-增益单圈有向图的谱。此外，我们分别通过零特征值的存在性，刻画了拉普拉斯矩阵和无符号拉普拉斯矩阵的圈平衡性与反平衡性。
英文摘要
A \(\mathbb{T}\)-gain digraph is a directed graph with complex unit gains on its arcs, allowing for no restrictions on oppositely directed arcs. This framework unifies various graph types, such as signed graphs, mixed graphs, complex unit gain graphs, digraphs, and signed digraphs. The gain adjacency, Laplacian, and signless Laplacian matrices are generally non-Hermitian with complex spectra. We develop the spectral theory of these matrices, extending classical results from signed digraphs and complex unit-gain graphs. For the gain adjacency matrix, we establish determinant and characteristic polynomial formulas, characterize cycle balance through switching equivalence and cospectrality with the underlying digraph. We further bound the spectral radius in terms of the underlying digraph and its maximum out-degree, with equality characterized by $μ$-balance. As a consequence, we determine the spectra of $\mathbb{T}$-gain unicyclic digraphs. Moreover, we characterize cycle balance and antibalance for the Laplacian and signless Laplacian matrices, respectively, through the presence of a zero eigenvalue.
