论文标题
平衡增益图的小组表示方法
A group representation approach to balance of gain graphs
论文作者
论文摘要
我们通过调查其邻接矩阵及其光谱来研究$ g $ gain图的余额，其中$ g $是任意组。作为第一步，我们表征了切换等价和增益图的平衡，以$ m_n（\ Mathbb c g）$中的邻接矩阵来表征。然后，我们通过扩展从组代数$ \ mathbb c g $将傅立叶变换的理论扩展到代数$ m_n（\ Mathbb c g）$，引入了与增益图和组表示相关的代表邻接矩阵。我们证明，只有当且仅当与任何（或等同于所有）忠实的统一表示与基础图的频谱相吻合时，增益图是平衡的，并且仅当具有由代表程度给出的多重性。我们表明，单位增益图的复杂邻接矩阵和封面图的邻接矩阵确实是我们结构的特殊情况。这使我们能够恢复一些经典的结果，并在这些图的光谱，索引或结构方面证明了一些新的平衡特征。
We study the balance of $G$-gain graphs, where $G$ is an arbitrary group, by investigating their adjacency matrices and their spectra. As a first step, we characterize switching equivalence and balance of gain graphs in terms of their adjacency matrices in $M_n(\mathbb C G)$. Then we introduce a represented adjacency matrix, associated with a gain graph and a group representation, by extending the theory of Fourier transforms from the group algebra $\mathbb C G$ to the algebra $M_n(\mathbb C G)$. We prove that a gain graph is balanced if and only if the spectrum of the represented adjacency matrix associated with any (or equivalently all) faithful unitary representation of $G$ coincides with the spectrum of the underlying graph, with multiplicity given by the degree of the representation. We show that the complex adjacency matrix of unit gain graphs and the adjacency matrix of a cover graph are indeed particular cases of our construction. This enables us to recover some classical results and prove some new characterizations of balance in terms of spectrum, index or structure of these graphs.
