Yuxuan Li
We establish a lower bound for the energy of a complex unit gain graph in terms of the matching number of its underlying graph, and characterize all the complex unit gain graphs whose energy reaches this bound.
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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 …
A complex unit gain graph (or ${\mathbb T}$-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 $\mathbb{T}= \{ z \in C:|z|=1 \}$ and a gain function $φ: \overrightarrow{E} \rightarrow \mathbb{T}$ with the property that $φ(e_{i,j})=φ(e_{j,i})^{-1}$. In this paper, we prove that $2m(G)-2c(G) \leq r(G, φ) \leq 2m(G)+c(G)$, where $r(G, φ)$, $m(G)$ and $c(G)$ are the rank of the Hermitian adjacency matrix $H(G, φ)$, the matching number and the cyclomatic number of $G$, respect…
Spectral Properties of Dual Unit Gain Graphs
本文在对偶单位增益图框架下研究对偶四元数和对偶复数单位增益图，建立交错定理、谱半径比较与平衡性刻画，并给出圈图邻接矩阵和拉普拉斯矩阵特征值的闭式表达及系数定理。
A bound on the girth of quaternion unit gain graphs in terms of the rank
该研究针对四元数单位增益图，得到其围长g关于邻接矩阵秩r的界g ≤ r + 2，刻画了满足g=r+2的图类，扩展了普通图、符号图及复单位增益图的相关结果。
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…
Spectral Properties of Complex Unit Gain Graphs
复数单位增益图是这样的图，其中边缘的每个方向被给予复数单位，该复数单位是分配给相反方向的复数单位的逆。我们将谱图论中的一些基本概念推广到复单位增益图。我们定义了邻接矩阵、关联矩阵和拉普拉斯矩阵，并对它们进行了研究。本文的主要结果是邻接矩阵和Laplacian矩阵的特征值界。
A complex unit gain graph is a simple graph in which each orientation of an edge is given a complex number with modulus 1 and its inverse is assigned to the opposite orientation of the edge. In this article, first we establish bounds for the eigenvalues of the complex unit gain graphs. Then we study some of the properties of the adjacency matrix of complex unit gain graph in connection with the characteristic and the permanental polynomials. Then we establish spectral properties of the adjacency matrices of complex unit gain graphs. In particular, using Perron-Frobenius theory, we establish a …
Gain distance Laplacian matrices for complex unit gain graphs
本文定义了复数单位增益图在任意顶点排序下的两种增益距离拉普拉斯矩阵，刻画了其奇异性与秩的公式，并利用该矩阵给出了图中平衡性的两类刻画，相关结果推广至加权情形。
Complex unit gain bicyclic graphs with rank 2, 3 or 4
$\mathbb{T}$增益图是由图$G=(V,E)$、圆组$\mathbb{T}=\{z\in C: |z|=1\}$和增益函数$φ:\overrightarrow{E}\rightarrow \mathbb{T}$组成的三重$Φ=(G,\mathbb{T},φ)$，使得$φ(e_{ij})=φ(e_{ji})^{-1}=\overline{φ(e_{ji})}$。用$r(Φ)$表示的$\mathbb{T}$ -增益图$Φ$的秩是$Φ$的邻接矩阵的秩。2015年，Yu，Qu和Tu [ G. H. Yu，H. Qu，J.H.涂，复单位增益图的惯性，应用数学计算。265（2015）619--629 ]获得了$\mathbb{T}$ -增益图的惯性的一些性质。他们刻画了具有小正指数或小负指数的$\mathbb{T}$ -增益单圈图。受此启发，本文刻画了秩为2，3，4的复单位增益双圈图。
Symmetry in complex unit gain graphs and their spectra
本文研究复单位增益图的结构、谱与符号对称性之间的关系，并构造无穷多个切换互异、谱对称但非符号对称的图，推广了符号图中的存在性结论。
A complex unit gain graph (or $\mathbb{T}$-gain graph) is a triple $Φ=(G, \mathbb{T}, φ)$ ($(G, φ)$ for short) consisting of a graph $G$ as the underlying graph of $(G, φ)$, $\mathbb{T}= \{ z \in C:|z|=1 \} $ is a subgroup of the multiplicative group of all nonzero complex numbers $\mathbb{C}^{\times}$ and a gain function $φ: \overrightarrow{E} \rightarrow \mathbb{T}$ such that $φ(e_{ij})=φ(e_{ji})^{-1}=\overline{φ(e_{ji})}$. In this paper, we investigate the relation among the rank, the independence number and the cyclomatic number of a complex unit gain graph $(G, φ)$ with order $n$, and pro…
Spectral properties of distance Laplacian matrices of complex unit gain graphs
该文研究复单位增益图的距离拉普拉斯矩阵谱性质，建立平衡增益图的零化度刻画，给出切换等价图同谱的必要条件，还推导了谱半径的上下界及等号成立条件。
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…
The row left rank of a quaternion unit gain graph in terms of maximum degree
本文研究四元数单位增益图的行左秩下界，证明了其至少为阶数除以最大度，并在连通情形下给出改进下界，同时刻画了所有极值图。
The left row rank of quaternion unit gain graphs in terms of girth
本文研究了四元数单位增益图的左行秩与围长的关系，证明了 $r(Φ)\geq g-2$，并刻画了满足特定等式及秩为 2 的图，推广了简单图、符号图和复单位增益图的相关结论。
Oriented Gain Graphs, Line Graphs and Eigenvalues
本文提出了增益图（电压图）上的定向理论，推广了图和符号图上的定向概念。利用这种定向方案，研究了增益图的线图。对于一个特殊的家庭增益图与复杂的单位，矩阵性质的建立。与图和符号图一样，在复数单位增益图的关联矩阵和线图的邻接矩阵之间存在关系。
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…
Bounds and extremal graphs for the energy of complex unit gain graphs
本文研究复单位增益图的能量，建立了连通无三角形图的下界与上界，并刻画极图，同时探讨单圈图上的最大最小增益能量。
The determinant of the Laplacian matrix of a quaternion unit gain graph
该研究针对四元数单位增益图的Laplacian矩阵行列式问题，利用新提出的行列非交换行列式给出了其组合描述，并通过数值例子验证了结果。
Let $Φ=(G, φ)$ be a complex unit gain graph (or $\mathbb{T}$-gain graph) and $A(Φ)$ be its adjacency matrix, where $G$ is called the underlying graph of $Φ$. The rank of $Φ$, denoted by $r(Φ)$, is the rank of $A(Φ)$. Denote by $θ(G)=|E(G)|-|V(G)|+ω(G)$ the dimension of cycle spaces of $G$, where $|E(G)|$, $|V(G)|$ and $ω(G)$ are the number of edges, the number of vertices and the number of connected components of $G$, respectively. In this paper, we investigate bounds for $r(Φ)$ in terms of $r(G)$, that is, $r(G)-2θ(G)\leq r(Φ)\leq r(G)+2θ(G)$, where $r(G)$ is the rank of $G$. As an applicatio…
On the Extremal Energy of Complex Unit Gain Dumbbell Graphs
研究以哑铃图 $D_{r,s,\ell}$ 为底图的复单位增益图的极值能量问题，通过切换等价将谱约化为两个环增益的实部，并利用匹配多项式得到特征多项式的显式表达式，最终确定了所有情况下的最大能量条件以及除 $r,s,\ell$ 全奇数外的最小能量条件。
The row left rank of quaternion unit gain graphs in terms of pendant vertices
本文研究四元数单位增益图的行左秩，利用悬挂顶点给出若干下界，并完整刻画了达到这些下界的极值图。
Characterization and equilibrium of bichromatic max-sum matchings
研究有限平面点集的双色最大和匹配及匹配平衡，证明最大和匹配与有向红圈增益的关系并得出最优性条件，刻画平衡匹配，揭示平衡与多种条件等价，在不同情形下有不同分类。
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…
Symmetry classes of spanning trees of Aztec diamonds and perfect matchings of odd squares with a unit hole
如果两个图的邻接矩阵是相似矩阵，我们称这两个图相似。我们发现，正方形网格$n$的顺序$n$是类似的不相交的工会的两个副本的四分之一阿兹特克钻石$QAD_{n-1}$的顺序$n-1$与路径$P_n^{(2)}$上的$n$顶点的边权重等于~2。我们的证明是基于一个明确的变化的基础上的向量空间的邻接矩阵的行为。证明这种基变换有效的论据是组合的。特别是，这允许计算四分之一阿兹特克钻石的生成树的数量。我们提出并分析了三个家庭的图形，共享上述“线性平方”属性的正方形网格：奇阿兹特克钻石，混合阿兹特克钻石，和阿兹特克枕套-图形从两个副本的阿兹特克钻石通过确定相应的顶点上的凸包。我们应用上面的结果来枚举所有的对称类的生成树的偶阿兹特克钻石，和所有的对称类不涉及旋转的生成树的奇和混合阿兹特克钻石。我们还列举了所有的，但基本情况下的对称类的完美匹配的奇方形网格与中心顶点删除。此外，我们得到了一个产品的阿兹特克枕套的生成树的数量公式。
On Spectra of $\mathbb{T}$-Gain Digraphs
该研究发展了$\boldsymbol{\top}$-增益有向图的增益邻接、拉普拉斯等矩阵的谱理论，扩展了相关经典结果，确定了单圈有向图的谱，并刻画了圈平衡等性质。
On the spectrum of complex unit gain graph
本文研究复单位增益图的谱性质，建立图是树的谱等价条件，刻画一类连通图上谱半径的界，并给出谱半径等于最大顶点度的增益图特征，推广了有向图 Hermitian 邻接矩阵的相关结果。
Applications of Graphical Condensation for Enumerating Matchings and Tilings
利用图凝聚技术证明了平面二部图的（完美）匹配数与区域分块数之间的各种组合恒等式。图形凝聚涉及将图的匹配叠加到较小子图的匹配上，然后以两种可能的方式之一将联合匹配（实际上是多重图）重新划分为另外两个子图的匹配。这种技术可以用来枚举各种各样的二部可平面图的完美匹配。应用程序包括阿兹特克钻石和矩形的多米诺骨牌平铺，堡垒的空竹平铺，平面分区和转置互补平面分区。
An edge-weighted, vertex-capacitated graph G is called stable if the value of a maximum-weight capacity-matching equals the value of a maximum-weight fractional capacity-matching. Stable graphs play a key role in characterizing the existence of stable solutions for popular combinatorial games that involve the structure of matchings in graphs, such as network bargaining games and cooperative matching games. The vertex-stabilizer problem asks to compute a minimum number of players to block (i.e., vertices of G to remove) in order to ensure stability for such games. The problem has been shown to …
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…
Optimal factor matchings for point processes on non-amenable unimodular graphs
本文研究了非可算图上点过程的最优因子匹配问题，证明在特定条件下最优匹配距离可通过确定性方案实现。
Matching complexes of $\bf 3 \times n$ grid graphs
本文研究了3×n网格图的匹配复形，证明其与多个球面的楔和同伦等价，并给出了楔中球面维度的完整列表。
In this paper we investigate the structure of flip graphs on non-crossing perfect matchings in the plane. Specifically, consider all non-crossing straight-line perfect matchings on a set of $2n$ points that are placed equidistantly on the unit circle. A flip operation on such a matching replaces two matching edges that span an empty quadrilateral with the other two edges of the quadrilateral, and the flip is called centered if the quadrilateral contains the center of the unit circle. The graph $\mathcal{G}_n$ has those matchings as vertices, and an edge between any two matchings that differ in…
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