A group representation approach to balance of gain graphs

Abstract

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 (CG). 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 CG to the algebra M n (CG). We prove that a gain graph is balanced if and only if the spectrum of the represented adjacency matrix associated with any (or equivalently all) faithful unitary representation of G 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 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.

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

“…In particular, the space M n×n (CG), or simply M n (CG), is also an algebra with involution * . For more details see [5].…”

“…When the representation is faithful, the matrix A Γ,ψ (π) contains all the information about the gain graph (Γ, ψ). For example, if the spectrum of A Γ,ψ (π) is known, then it is possible to establish whether (Γ, ψ) is balanced or not (see [5]).…”

“…In the recent years many works were devoted to the investigation, in the setting of gain graphs, of several topics coming from the classical case. Among them, there are graph matrices and their spectra (see [3,5,10,[12][13][14]), or the concept of orientation and line graph (see [2,13,20]). All of them are inspired by the corresponding generalizations existing for signed graphs [1,4,15,17,21].…”

“…The main issue is that a gain graph (Γ, ψ) does not have a canonical complex adjacency matrix. A solution comes from the matrix Fourier transform approach, already fruitful in [5]. Suppose that Γ has n vertices and π is a unitary representation of G of degree k; we can define the represented adjacency matrix A Γ,ψ (π) ∈ M nk (C).…”

Graph covers with two new eigenvalues

“…This study has blossomed in many different directions whose references are recorded by Zaslavsky in the dynamic bibliography [24]. Last month a new paper of Cavaleri, D'Angeli, and Donno [6] was added to the arxiv, wherein the authors answer a question of Zaslavsky on the algebraic characterization of balanced gain graphs. In the final section of this paper, the authors call upon the connection between gain graphs and covering graphs to translate their results to the latter setting.…”