Cycle and circle tests of balance in gain graphs: Forbidden minors and their groups

Rybnikov, Konstantin; Zaslavsky, Thomas

doi:10.1002/jgt.20116

Cited by 5 publications

(4 citation statements)

References 8 publications

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“…For a given group G, a G-gain graph is a graph G with each orientation of an edge of G is assigned an element g ∈ G (called gain of the oriented edge) and whose inverse g −1 is assigned to the opposite orientation of the edge. For more details about the notion of G-gain graphs, we refer to [14,15,16,17,18,19]. Let T = {z ∈ C : |z| = 1} be the multiplicative group of unit complex numbers.…”

Section: Introductionmentioning

confidence: 99%

On the spectrum of complex unit gain graph

Samanta¹,

Kannan²

2019

Preprint

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A T-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is called T-gain adjacency matrix. Let T G denote the collection of all T-gain adjacency matrices on a graph G. In this article, we study the cospectrality of matrices in T G and we establish equivalent conditions for a graph G to be a tree in terms of the spectrum and the spectral radius of matrices in T G . We identify a class of connected graphs F such that for each G ∈ F , the matrices in T G have nonnegative real part up to diagonal unitary similarity. Then we establish bounds for the spectral radius of T-gain adjacency matrices on G ∈ F in terms of their largest eigenvalues. Thereupon, we characterize T-gain graphs for which the spectral radius of the associated T-gain adjacency matrices equal to the largest vertex degree of the underlying graph. These bounds generalize results known for the spectral radius of Hermitian adjacency matrices of digraphs and provide an alternate proof of a result about the sharpness of the bound in terms of largest vertex degree established in [Krystal Guo, Bojan Mohar. Hermitian adjacency matrix of digraphs and mixed graphs.

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Section: Introductionmentioning

confidence: 99%

On the spectrum of complex unit gain graph

Samanta¹,

Kannan²

2019

Preprint

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“…Actually it is not necessary to check the condition on every closed walk, but only on the simple ones (without repetitions of vertices, that are finitely many). Even more, it is enough to test the gain only on a fundamental system of circles, see [27, Corollary 3.2] and [18,19] for further developments in this direction.…”

Section: Gain Graphs and Balancementioning

confidence: 99%

A group representation approach to balance of gain graphs

Cavaleri,

D'Angeli,

Donno

2020

Preprint

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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.

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“…There are also finite counterexamples, but the gain group necessarily has an element of odd order. Two such examples, based respectively on the wheel and on an even circle with all edges doubled, are found in [19,Theorem 6.16]. …”

Section: Fig 1 Counterexample To the Circle Test With Infinitely 2-mentioning

confidence: 99%

“…We also show an infinite graph that fails the Binary Cycle Test for all gain groups that have an infinitely 2-divisible element other than the identity. (In [19] we prove that for any gain group with non-trivial elements of odd order there is a graph for which the Binary Cycle Test does not work.…”

Section: Introductionmentioning

confidence: 99%

Criteria for Balance in Abelian Gain Graphs, with Applications to Piecewise-Linear Geometry

Rybnikov

Zaslavsky

2005

Discrete Comput Geom

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Consider a gain graph with abelian gain group having no odd torsion. If there is a basis of the graph's binary cycle space, each of whose members can be lifted to a closed walk whose gain is the identity, then the gain graph is balanced, provided that the graph is finite or the group has no non-trivial infinitely 2-divisible elements. We apply this theorem to deduce a result on the projective geometry of piecewise-linear realizations of cell-decompositions of manifolds.

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Cycle and circle tests of balance in gain graphs: Forbidden minors and their groups

Cited by 5 publications

References 8 publications

On the spectrum of complex unit gain graph

On the spectrum of complex unit gain graph

A group representation approach to balance of gain graphs

Criteria for Balance in Abelian Gain Graphs, with Applications to Piecewise-Linear Geometry

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