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. The energy $ \mathcal{E}(Φ) $ of a $ \mathbb{T} $-gain graph $ Φ$ is the sum of the absolute values of all eigenvalues of $ A(Φ) $.
We study the notion of energy of a vertex of a $ \mathbb{T} $-gain graph, and establish bounds for it. For any $ \mathbb{T} $-gain graph $ Φ$, we prove that $2τ(G)-2c(G) \leq \mathcal{E}(Φ) \leq 2τ(G)\sqrt{Δ(G)}$, where $ τ(G), c(G)$ and $ Δ(G)$ are the vertex cover number, the number of odd cycles and the largest vertex degree of $ G $, respectively. Furthermore, using the properties of vertex energy, we characterize the classes of $ \mathbb{T} $-gain graphs for which $ \mathcal{E}(Φ)=2τ(G)-2c(G) $ holds. Also, we characterize the classes of $ \mathbb{T} $-gain graphs for which $\mathcal{E}(Φ)= 2τ(G)\sqrt{Δ(G)} $ holds. This characterization solves a general version of an open problem. In addition, we establish bounds for the energy in terms of the spectral radius of the associated adjacency matrix.
A complex unit gain graph ($ \mathbb{T} $-gain graph), $ Φ=(G, φ) $ is a graph where the function $ φ$ assigns a unit complex number to each orientation of an edge of $ G $, and its inverse is assigned to the opposite orientation. %A complex unit gain graph($ \mathbb{T} $-gain graph) is a simple graph where each orientation of an edge is given a complex unit, and its inverse is assigned to the opposite orientation of the edge. In this article, we propose gain distance matrices for $ \mathbb{T} $-gain graphs. These notions generalize the corresponding known concepts of distance matrices and sig…
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 graph to attain this minimum. This allows us to draw a surprising parallel to well-studied systems of lines in complex space, through a natural correspondence to unit-norm tight frames. We offer a full classification of two-eigenvalue gain graphs wit…
A complex unit gain graph is a triple $φ=(G, \mathbb{T}, φ)$ (or $G^φ$ for short) consisting of a simple graph $G$, as the underlying graph of $G^φ$, the set of unit complex numbers $\mathbb{T}={z\in \mathbb{C}: |z| = 1}$ and a gain function $φ: \overrightarrow{E}\rightarrow \mathbb{T}$ such that $φ(e_{i,j})=φ(e_{j,i}) ^{-1}$. Let $A(G^φ)$ be adjacency matrix of $G^φ$. In this paper, we prove that $$m(G)-c(G)\leq p(G^φ)\leq m(G)+c(G),$$ $$m(G)-c(G)\leq n(G^φ)\leq m(G)+c(G),$$ where $p(G^φ)$, $n(G^φ)$, $m(G)$ and $c(G)$ are the number of positive eigenvalues of $A(G^φ)$, the number of negative …
Let $ Φ=(G, φ) $ be a connected complex unit gain graph ($ \mathbb{T} $-gain graph) on a simple graph $ G $ with $ n $ vertices and maximum vertex degree $ Δ$. The associated adjacency matrix and degree matrix are denoted by $ A(Φ) $ and $ D(Φ) $, respectively. Let $ m_α(Φ,λ) $ be the multiplicity of $ λ$ as an eigenvalue of $ A_α(Φ) :=αD(Φ)+(1-α)A(Φ)$, for $ α\in[0,1) $. In this article, we establish that $ m_α(Φ, λ)\leq \frac{(Δ-2)n+2}{Δ-1}$, and characterize the classes of graphs for which the equality hold. Furthermore, we establish a couple of bounds for the rank of $A(Φ)$ in terms of the…
On the Extremal Energy of Complex Unit Gain Dumbbell Graphs
College of Information Science and Technology, Jinan University（济南大学信息科学与技术学院）；Departamento de Matemática, Universidad Nacional de San Luis（圣路易斯国立大学数学系）
We introduce a switching operation, inspired by the Godsil-McKay switching, in order to obtain pairs of $G$-cospectral gain graphs, that are gain graphs cospectral with respect to every representation of the gain group $G$. For instance, for two signed graphs, this notion of cospectrality is equivalent to the cospectrality of their signed adjacency matrices together with the cospectrality of their underlying graphs. Moreover, we introduce another more flexible switching in order to obtain pairs of gain graphs cospectral with respect to some fixed unitary representation. Many existing notions o…
A complex unit gain graph ($\mathbb{T}$-gain graph), $Φ= (G, φ)$ is a graph where the function $φ$ assigns a unit complex number to each orientation of an edge of $G$, and its inverse is assigned to the opposite orientation. A $ \mathbb{T} $-gain graph $Φ$ is balanced if the product of the edge gains of each cycle (with a fixed orientation) is $1$. Signed graphs are special cases of $\mathbb{T}$-gain graphs. The adjacency matrix of $Φ$, denoted by $ \mathbf{A}(Φ)$ is defined canonically. The gain Laplacian for $Φ$ is defined as $\mathbf{L}(Φ) = \mathbf{D}(Φ) - \mathbf{A}(Φ)$, where $\mathbf{D}…
A group representation approach to balance of gain graphs
通过研究邻接矩阵和谱，研究了$G$ -增益图的平衡性，其中$G$是任意群.作为第一步，我们在$M_n(\mathbb C G)$的邻接矩阵的切换等价性和平衡的增益图的特征。然后，我们介绍了一个表示邻接矩阵，与增益图和群表示，通过扩展的傅里叶变换理论从群代数$\mathbb C G$到代数$M_n(\mathbb C G)$。我们证明了一个增益图是平衡的当且仅当频谱的表示邻接矩阵与任何（或等价的所有）忠实酉表示$G$与频谱的基础图相一致，与多重度的表示。我们证明了单位增益图的复邻接矩阵和覆盖图的邻接矩阵确实是我们构造的特例。这使我们能够恢复一些经典的结果，并证明了一些新的性质的平衡，这些图的谱，指数或结构。
