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, anytime G admits a finite-dimensional faithful unitary representation \(\pi \), a G-gain graph is balanced if and only if the spectrum of the represented adjacency matrix associated with \(\pi \) 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 complex 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.

通过研究它们的邻接矩阵和光谱，我们研究了G增益图的平衡，其中G是任意组。第一步，我们根据\（M_n（\ mathbb CG）\）中的邻接矩阵来描述增益图的切换等价关系和平衡。然后，通过将傅立叶变换的理论从群代数\（\ mathbb CG \）扩展到代数\（M_n（\ mathbb CG）\），引入与增益图和组表示相关联的表示的邻接矩阵。我们证明，任何时候ģ承认有限维忠实酉表示\（\ PI \） ，一个ģ当且仅当与\（\ pi \）关联的表示的邻接矩阵的频谱与基础图的频谱重合时，-增益图才是平衡的，并且多样性由表示的程度给出。我们表明，复杂单位增益图的复杂邻接矩阵和覆盖图的邻接矩阵确实是我们构造的特殊情况。这使我们能够恢复一些经典结果，并根据这些图的频谱，索引或结构证明平衡的一些新特性。