Back to Search Start Over

A group representation approach to balance of gain graphs

A group representation approach to balance of gain graphs

Authors :
Daniele D'Angeli
Matteo Cavaleri
Alfredo Donno
Publication Year :
2020
Publisher :
arXiv, 2020.

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 $M_n(\mathbb C G)$. 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 $\mathbb C G$ to the algebra $M_n(\mathbb C G)$. 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.<br />Comment: 27 pages, 3 tables, 5 figures. In this second version, the word "balancedness" (with the meaning of "property of being balanced") has been replaced by the word "balance" both in the title and in the body of the article

Details

Database :
OpenAIRE
Accession number :
edsair.doi.dedup.....5fc6a90d57c16f71fb465be4877905d2
Full Text :
https://doi.org/10.48550/arxiv.2001.08490