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

TL;DR

This paper introduces a generalized framework for gain-line graphs using $G$-phases and group representations, extending previous abelian results to arbitrary groups and exploring spectral properties.

Contribution

It defines gain-line graphs via $G$-phases, proves invariance of switching classes, and characterizes gain functions and their properties using group algebra and spectral methods.

Findings

Switching equivalence class of gain functions is invariant under different $G$-phases.

Provides spectral conditions for a gain graph to be a gain-line graph.

Generalizes previous abelian results to arbitrary groups.

Abstract

Let $G$ be an arbitrary group. We define a gain-line graph for a gain graph $(\Gamma,\psi)$ through the choice of an incidence $G$-phase matrix inducing $\psi$. We prove that the switching equivalence class of the gain function on the line graph $L(\Gamma)$ does not change if one chooses a different $G$-phase inducing $\psi$ or a different representative of the switching equivalence class of $\psi$. 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 $\mathcal H_\Gamma$ of $G$-phases of $\Gamma$ allows us to characterize gain functions on $\Gamma$, gain functions on $L(\Gamma)$, 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.