论文标题
复杂单元增益图的能量的边界
Bounds for the energy of a complex unit gain graph
论文作者
论文摘要
$ \ mathbb {t} $ - 增益图，$φ=（g，φ）$，是一个图形，其中函数$φ$将单位复数编号分配给边缘的每个方向，并且其逆向分配给相反的方向。相关的邻接矩阵$ a（φ）$是按典型定义的。 $ \ mathbb {t} $ gain Graph $φ$的能量$ \ MATHCAL {E}（φ）$是$ a（φ）$的所有特征值的绝对值的总和。 我们研究了$ \ mathbb {t} $ gain图的顶点的能量概念，并为其建立界限。对于任何$ \ MATHBB {t} $ gain graph $φ$，我们证明$2τ（g）-2c（g）\ leq \ leq \ mathcal {e}（φ）\ leq2τ（g）\ sqrt \ sqrt {δ（g）}周期和最大的$ g $顶点学位。此外，使用顶点能量的属性，我们表征了$ \ mathbb {t} $ gain-gain graphs的类，其中$ \ mathcal {e}（φ）=2τ（g）-2c（g）-2c（g）$ holds。另外，我们表征了$ \ mathbb {t} $ gain graphs的类别，其中$ \ mathcal {e}（φ）=2τ（g）\ sqrt {δ（g）} $保持。此特征解决了开放问题的一般版本。此外，我们根据相关邻接矩阵的光谱半径建立了能量的界限。
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.
