Gain-line graphs via $G$-phases and group representations

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.

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