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Eigenvalues of complex unit gain graphs and gain regularity
Special MatricesA complex unit gain graph (or T{\mathbb{T}}-gain graph) Γ=(G,γ)\Gamma =\left(G,\gamma ) is a gain graph with gains in T{\mathbb{T}}, the multiplicative group of complex units.
Brunetti Maurizio
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Balancedness and the Least Laplacian Eigenvalue of Some Complex Unit Gain Graphs
Discussiones Mathematicae Graph Theory, 2020 Let 𝕋4 = {±1, ±i} be the subgroup of 4-th roots of unity inside 𝕋, the multiplicative group of complex units. A complex unit gain graph Φ is a simple graph Γ = (V (Γ) = {v1, . . .
Belardo Francesco +2 more
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Spectral properties of complex unit gain graphs [PDF]
Linear Algebra and Its Applications, 2012 13 pages, 1 figure, to appear in Linear Algebra ...
Nathan Reff
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Unit gain graphs with two distinct eigenvalues and systems of lines in complex space
Discrete Mathematics, 2022 Since the introduction of the Hermitian adjacency matrix for digraphs, interest in so-called complex unit gain graphs has surged. In this work, we consider gain graphs whose spectra contain the minimum number of two distinct eigenvalues. Analogously to graphs with few distinct eigenvalues, a great deal of structural symmetry is required for a gain ...
Edwin Van Dam
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Symmetry in complex unit gain graphs and their spectra
Linear Algebra and Its ApplicationsComplex unit gain graphs may exhibit various kinds of symmetry. In this work, we explore structural symmetry, spectral symmetry and sign-symmetry in such graphs, and their respective relations to one-another. Our main result is a construction that transforms an arbitrary complex unit gain graph into infinitely many switching-distinct ones whose ...
Edwin Van Dam
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Bounds and extremal graphs for the energy of complex unit gain graphs
Linear Algebra and Its ApplicationsA complex unit gain graph ($ \mathbb{T} $-gain graph), $ Φ=(G, φ) $ is a graph where the gain function $ φ$ assigns a unit complex number to each orientation of an edge of $ G $ and its inverse is assigned to the opposite orientation. The associated adjacency matrix $ A(Φ) $ is defined canonically. The energy $ \mathcal{E}(Φ) $ of a $ \mathbb{T} $-gain
Rajesh Kannan M, Aniruddha Samanta
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Bounds for the energy of a complex unit gain graph [PDF]
Linear Algebra and its Applications, 2021 A $\mathbb{T}$-gain graph, $Φ= (G, φ)$, is a graph in which the function $φ$ assigns a unit complex number to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix $ A(Φ) $ is defined canonically.
Aniruddha Samanta, M. Rajesh Kannan
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On the spectrum of complex unit gain graph
Discrete Mathematics, 2019 A $\mathbb{T}$-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 $\mathbb{T}$-gain adjacency matrix.
Samanta, Aniruddha, Kannan, M. Rajesh
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NEPS of complex unit gain graphs
The Electronic Journal of Linear Algebra, 2023 A complex unit gain graph (or $\mathbb T$-gain graph) is a gain graph with gains in $\mathbb T$, the multiplicative group of complex units. Extending a classical construction for simple graphs due to Cvektovic, suitably defined noncomplete extended $p$-sums (NEPS, for short) of $\mathbb T$-gain graphs are considered in this paper. Structural properties
Francesco Belardo +2 more
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Line graphs of complex unit gain graphs with least eigenvalue -2
The Electronic Journal of Linear Algebra, 2021 Let $\mathbb T$ be the multiplicative group of complex units, and let $\mathcal L (\Phi)$ denote a line graph of a $\mathbb{T}$-gain graph $\Phi$. Similarly to what happens in the context of signed graphs, the real number $\min Spec(A(\mathcal L (\Phi))$, that is, the smallest eigenvalue of the adjacency matrix of $\mathcal L(\Phi)$, is not less than $-
Belardo F., Brunetti M.
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