Gain-line graphs via G-phases and group representations

Let $G$ be an arbitrary group. We define a gain-line graph for gain $(\Gamma,\psi)$ through the choice of incidence $G$-phase matrix inducing $\psi$. prove that switching equivalence class function on line $L(\Gamma)$ does not change if one chooses different $\psi$ or representative In this way, we generalize to any group some results proven by N. Reff in abelian case. The investigation orbits natural actions set $\mathcal H_\Gamma$ $G$-phases $\Gamma$ allows us characterize functions $\Gamma$, $L(\Gamma)$, their classes and balance property. use algebra valued matrices plays fundamental role and, together with Fourier transform, represent Hermitian perform spectral computations. Our also provide necessary conditions graph.