论文标题
通过$ g $ pluned和组表示的增益线图
Gain-line graphs via $G$-phases and group representations
论文作者
论文摘要
令$ g $为任意组。我们通过选择$ g $ ch $ - 相矩阵诱导$ψ$来定义增益图$（γ，ψ）$的增益线图。我们证明，如果选择了$ψ$的不同$ g $ - 相或其他代表$ψ$的$ g $ - 相位，则在line Graph $ l（γ）$上的开关等价类别不会更改。通过这种方式，我们将N. Reff在Abelian案中证明的任何群体都概括了一些结果。对$ g $ $ g $的$ g $的某些自然动作的轨道的调查，$ g $的$ g $ - $γ$的浓度使我们能够表征$γ$上的增益功能，在$ l（γ）$，其开关等价类别上的增益功能，其开关等价类别及其平衡财产。组代数估算矩阵的使用起着基本作用，并且与矩阵傅立叶变换一起，使我们能够用Hermitian矩阵代表增益图并执行光谱计算。我们的光谱结果还为增益图成为增益线图提供了一些必要的条件。
Let $G$ be an arbitrary group. We define a gain-line graph for a gain graph $(Γ,ψ)$ through the choice of an incidence $G$-phase matrix inducing $ψ$. We prove that the switching equivalence class of the gain function on the line graph $L(Γ)$ does not change if one chooses a different $G$-phase inducing $ψ$ or a different representative of the switching equivalence class of $ψ$. In this way, we generalize to any group some results proven by N. Reff in the abelian case. The investigation of the orbits of some natural actions of $G$ on the set $\mathcal H_Γ$ of $G$-phases of $Γ$ allows us to characterize gain functions on $Γ$, gain functions on $L(Γ)$, their switching equivalence classes and their balance property. The use of group algebra valued matrices plays a fundamental role and, together with the matrix Fourier transform, allows us to represent a gain graph with Hermitian matrices and to perform spectral computations. Our spectral results also provide some necessary conditions for a gain graph to be a gain-line graph.
