Weige Xi, Ligong Wang, Xiangxiang Liu, Xihe Li, Xiaoguo Tian
Let $λ_{2}(G)$ be the second smallest normalized Laplacian eigenvalue of a graph $G$. In this paper, we determine all unicyclic graphs of order $n\geq21$ with $λ_{2}(G)\geq 1-\frac{\sqrt{6}}{3}$. Moreover, the unicyclic graphs with $λ_{2}(G)=1-\frac{\sqrt{6}}{3}$ are also determined.
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For a graph $G$, the signless Laplacian Estrada index is defined as $SLEE(G)=\sum^{n}_{i=1}e^{q^{}_i}$,where $q_1, q_2, \dots, q_n$are the eigenvalues of the signless Laplacian matrix of $G$. In this paper, we first characterize the unicyclic graphs with the first two largest and smallest $SLEE$ and then determine the unique unicyclic graph with maximum $SLEE$ among the unicyclic graphs on $n$ vertices with given diameter.
Laplacian eigenvalue distribution for unicyclic graphs
该研究针对单圈图，以图的直径和围长为参数，给出了区间[0,1)内其拉普拉斯特征值数量的上界，推进了单圈图谱性质相关研究。
For a connected graph $G$, the distance Laplacian spectral radius of $G$ is the spectral radius of its distance Laplacian matrix $\mathcal{L}(G)$ defined as $\mathcal{L}(G)=Tr(G)-D(G)$, where $Tr(G)$ is a diagonal matrix of vertex transmissions of $G$ and $D(G)$ is the distance matrix of $G$. In this paper, we determine the unique graphs with maximum distance Laplacian spectral radius among unicyclic graphs.
Minimizing Laplacian spectral radius of unicyclic graphs with fixed girth
在本文中，我们考虑以下问题：在所有的简单连通单圈图类的$n$顶点与周长$g$（$n,g$是固定的），哪一个图最小化的拉普拉斯谱半径？我们证明了图$U_{n,g}$（定义在第1节）唯一地最小化拉普拉斯谱半径时，$g$是偶数的$n\geq 2g-1$和$g$是奇数的$n\geq 3g-1$.
On signless Laplacian coefficients of unicyclic graphs with given matching number
设$G$是$n$阶单圈图，$Q_G(x)= det(xI-Q(G))={matrix} \sum_{i=1}^n (-1)^i φ_i x^{n-i}{matrix}$是图$G$的无符号拉普拉斯矩阵的特征多项式.本文给出了$G$的一些变换，这些变换使集合$\mathcal{G}(n,m)$中的所有无符号拉普拉斯系数都减少. $\mathcal{G}(n,m)$表示所有具有匹配数$m$的n-顶点单圈图。我们刻画了使集合$\mathcal{G}(n,m)$中的所有无符号拉普拉斯系数最小的图.偶数）周长。此外，我们还在所有具有奇数（或奇数）的$n$ -顶点单圈图的集合$\mathcal{G}(n)$中找到了具有最小无符号Laplacian系数的极图.偶数）周长。
Laplacian coefficients of unicyclic graphs with the number of leaves and girth
设$G$是$n$阶图，$\mathcal{L}(G,λ)=\sum_{k=0}^n (-1)^{k}c_{k}(G)λ^{n-k}$是其Laplacian矩阵的特征多项式.由Ilić和Ilić猜想[A. Ilić，M. Ilić，Laplacian coefficients of trees with given number of leaves or vertices of degree two，线性代数及其应用431（2009）2195-2202.]本文研究了所有叶数为$l$的$n$ -顶点单圈图的集合$\mathcal{U}_{n,l}$中所有Laplacian系数最小的极图，其中$\mathcal{U}_{n,l}^g$表示叶数为$l$且围长为$g$的$n$ -顶点单圈图的集合$(\mathcal{U}_{n,l}^g, \preceq)$中极小元的性质。这些结果被用来反驳他们的猜想。此外，还研究了$\mathcal{U}_{n,l}^g$中具有最小类拉普拉斯能量的图。
Edge-grafting theorems on permanents of the Laplacian matrices of graphs and their applications
研究了具有最小Laplacian永久式的$n$顶点上的树和单圈图.本文利用边嫁接变换，确定了具有第二和第三小Laplacian积和式的给定二分划的$n$ -顶点树.同样地，给出了具有第一、第二和第三小Laplacian永久式的给定二分划的$n$ -顶点二部单圈图的刻画.从而确定了具有第一、第二和第三小Laplacian永久式的$n$ -顶点二部单圈图。
It has been an open problem to find the Moore-Penrose inverses of the incidence, Laplacian, and signless Laplacian matrices of families of graphs except trees and unicyclic graphs. Since the inverse formulas for an odd unicyclic graph and an even unicyclic graph are quite different, we consider wheel graphs as they are formed from odd or even cycles. In this article solve the open problem for wheel graphs. This work has an interesting connection to inverses of circulant matrices.
Unicyclic Graphs with equal Laplacian Energy
本文在一类具有输入图和输出图的拉普拉斯特征值相关性质的图上引入了一种新的运算。在此基础上，我们得到了一类具有相同Laplacian能量的n阶（n的平方根）非共谱单圈图。
The $α$-spectral radius of a connected graph $G$ is the spectral radius of $A_α$-matrix of $G$. In this paper, we discuss the methods for comparing $α$-spectral radius of graphs. As applications, we characterize the graphs with the maximal $α$-spectral radius among all unicyclic and bicyclic graphs of order $n$ with diameter $d$, respectively. Finally, we determine the unique graph with maximal signless Laplacian spectral radius among bicyclic graphs of order $n$ with diameter $d$.
Given a graph $G$, the burning number of $G$ is the smallest integer $k$ for which there are vertices $x_1, x_2,\ldots,x_k$ such that $(x_1,x_2,\ldots,x_k)$ is a burning sequence of $G$. It has been shown that the graph burning problem is NP-complete, even for trees with maximum degree three, or linear forests. A $t$-unicyclic graph is a unicycle graph with exactly one vertex of degree greater than $2$. In this paper, we first present the bounds for the burning number of $t$-unicyclic graphs, and then use the burning numbers of linear forests with at most three components to determine the burn…
Acyclic and unicyclic graphs whose minimum skew rank is equal to the minimum skew rank of a diametrical path
实数域上的单图G的最小斜秩是指当{i，j}是G中的边时，（i，j）-元（i不等于j）不为零，否则为零的所有实斜对称矩阵的最小可能秩.本文给出了一个计算连通单圈图的最小斜秩的算法，并对所有连通无圈图和连通单圈图G进行了分类，使G的最小斜秩等于P的最小斜秩，其中P是G的一条直径路。
The metric (resp. edge metric) dimension of a simple connected graph $G$, denoted by dim$(G)$ (resp. edim$(G)$), is the cardinality of a smallest vertex subset $S\subseteq V(G)$ for which every two distinct vertices (resp. edges) in $G$ have distinct distances to a vertex of $S$. It is an interesting topic to discuss the relation between dim$(G)$ and edim$(G)$ for some class of graphs $G$. In this paper, we settle two open problems on this topic for a widely studied class of graphs, called unicyclic graphs. Specifically, we introduce four classes of subgraphs to characterize the structure of a…
F-index of a graph is the sum of the cube of the degrees of the vertices. In this paper, we investigate the F-indices of unicyclic graphs by introducing some transformation, and characterize the unicyclic graphs with the first five largest F-indices and the unicyclic graphs with the first two smallest F-indices, respectively.
In this paper, we characterize all connected graphs with exactly three distinct normalized Laplacian eigenvalues of which one is equal to $1$, determine all connected bipartite graphs with at least one vertex of degree $1$ having exactly four distinct normalized Laplacian eigenvalues, and find all unicyclic graphs with three or four distinct normalized Laplacian eigenvalues.
The signless Laplacian Q and signless edge-Laplacian S of a given graph may or may not be invertible. The Moore-Penrose inverses of Q and S are studied. In particular, using the incidence matrix, we find combinatorial formulas of the Moore- Penrose inverses of Q and S for trees. Also we present combinatorial formulas of the inverses of Q and S for odd unicyclic graphs.
On the Main Signless Laplacian Eigenvalues of a Graph
一个图$G$的无符号拉普拉斯特征值称为主无符号拉普拉斯特征值，如果它有一个特征向量的元素之和不等于零.本文首先给出了图具有一个或两个主无符号Laplacian特征值的充要条件，然后分别刻画了树和单圈图具有恰好两个主无符号Laplacian特征值.
The independence polynomial of a graph $G$, denoted $I(G,x)$, is the generating polynomial for the number of independent sets of each size. The roots of $I(G,x)$ are called the \textit{independence roots} of $G$. It is known that for every graph $G$, the independence root of smallest modulus, denoted $ξ(G)$, is real. The relation $\preceq$ on the set of all graphs is defined as follows, $H\preceq G$ if and only if $I(H,x)\ge I(G,x)\text{ for all }x\in [ξ(G),0].$ We find the maximum and minimum connected unicyclic and connected well-covered unicyclic graphs of a given order with respect to $\pr…
A $k$-cyclic graph is a connected graph of order $n$ and size $n+k-1$. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all $C_{4}$-free $k$-cyclic graphs of order $n$. Furthermore, we determine the first three unicyclic, and bicyclic, $C_{4}$-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the (combinatorial) Laplacian.
In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) (resp. E(G)) is called the vertex (resp. edge) metric dimension of G. In [16] it was shown that both vertex and edge metric dimension of a unicyclic graph G always take values from just two explicitly given consecutive integers that are derived from the structure of the graph. A natural problem that arises is to determine under what conditions these dimensions take each of the two possible values. In this paper for each of these two metric dimensions we characterize three graph confi…
On the unicyclic graphs having vertices that belong to all their (strong) metric bases
本文研究单圈图的（强）基迫顶点问题，给出基于基迫顶点数量的单圈图分类结果，并证明单圈图可含任意多强基迫顶点及相关存在性刻画。
A spanning tree of a graph is a connected subgraph on all vertices with the minimum number of edges. The number of spanning trees in a graph $G$ is given by Matrix Tree Theorem in terms of principal minors of Laplacian matrix of $G$. We show a similar combinatorial interpretation for principal minors of signless Laplacian $Q$. We also prove that the number of odd cycles in $G$ is less than or equal to $\frac{\det(Q)}{4}$, where the equality holds if and only if $G$ is a bipartite graph or an odd-unicyclic graph.
The graph entropy describes the structural information of graph. Motivated by the definition of graph entropy in general graphs, the graph entropy of hypergraphs based on Laplacian degree are defined. Some results on graph entropy of simple graphs are extended to k-uniform hypergraphs. Using an edge-moving operation, the maximum and minimum graph entropy based on Laplacian degrees are determined in k-uniform hypertrees, unicyclic k-uniform hypergraphs, bicyclic k-uniform hypergraphs and k-uniform chemical hypertrees, respectively, and the corresponding extremal graphs are determined.
The generalized Ramsey number $R(H, K)$ is the smallest positive integer $n$ such that for any graph $G$ with $n$ vertices either $G$ contains $H$ as a subgraph or its complement $\overline{G}$ contains $K$ as a subgraph. Let $T_n$ be a tree with $n$ vertices and $F_m$ be a fan with $2m + 1$ vertices consisting of $m$ triangles sharing a common vertex. We prove a conjecture of Zhang, Broersma and Chen for $m \ge 9$ that $R(T_n, F_m) = 2n - 1$ for all $n \ge m^2 - m + 1$. Zhang, Broersma and Chen showed that $R(S_n, F_m) \ge 2n$ for $n \le m^2 -m$ where $S_n$ is a star on $n$ vertices, implying…
Characterizing the second smallest eigenvalue of the normalized Laplacian of a tree
本文证明了图的正规拉普拉斯算子λ 1的调和本征函数在图的铰接点上的单调性定理。本文给出了图的正规拉普拉斯矩阵的Perron分量的定义，并给出了它的次小特征值的刻画。
On the Smallest Eigenvalue of Grounded Laplacian Matrices
我们提供了上界和下界上的最小特征值的接地拉普拉斯矩阵（这是通过删除特定的行和列的拉普拉斯矩阵的给定图获得的矩阵）。上界和下界之间的间隙取决于与接地拉普拉斯算子的最小特征值相对应的特征向量的最小和最大分量的比率。我们提供了一个图论上的这个比率的界限，并随后获得了严格的特征值的最小特征值的某些类别的图。具体来说，对于Erdos Renyi随机图，我们表明，当一个（足够小）集$S$的行和列从拉普拉斯算子，并添加一个边缘的概率$p$是足够大的，接地拉普拉斯算子的最小特征值渐近几乎肯定接近$|S|p$。我们还表明，随机$d$ -正则图与一个单一的行和列删除，最小的特征值是$Θ(\frac{d}{n})$。我们的边界有应用程序的收敛速度在连续时间和离散时间的一致性动力学与顽固或领导节点的研究。
The smallest normalized signless $\infty$-Laplacian eigenvalue for non-bipartite connected graphs
本文研究了非二部连通图的最小规范化无符号∞-拉普拉斯特征值，证明其等于加权无符号关联矩阵广义逆的最小∞-范数的倒数，并举例说明。
Let $G$ be a simple connected graph with order $n$. Let $\mathcal{L}(G)$ be the normalized Laplacian matrix of $G$. Let $λ_{k}(G)$ be the $k$-th smallest normalized Laplacian eigenvalue of $G$. Denote $ρ(A)$ the spectral radius of the matrix $A$. In this paper, we study the behaviors of $λ_{2}(G)$ and $ρ(\mathcal{L}(G))$ when the graph is perturbed by three operations.
If $G$ is a graph, its Laplacian is the difference between diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs $G_{1}$ and $G_{2}$ is a graph $G=G_{1}\odot G_{2}$ with $V(G)=V(G_{1})\cup V(G_{2})$ and $E(G)= E(G_{1})\cup E(G_{2})\cup \{e=uv\}$ where $u\in V(G_1)$ and $v\in V(G_2)$. In this paper, we consider the eigenvector of unicycle graphs. We study the relationship between the Laplacian eigenvalue $2$ of unicyclic graphs $G_1$ and $G_2$; and bicyclic graphs $G=G_{1}\odot G_{2}$. We also characterize the broken sun graphs and the one edge conn…
On a conjecture for the signless Laplacian eigenvalues
设$G$是一个具有$n$顶点和$e(G)$边的简单图，$q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)\geq0$是$G.$的无符号Laplacian特征值。设$S_k^+(G)=\sum_{i=1}^{k}q_i(G),$，其中$k=1, 2, \ldots, n.$ F. Ashraf等人证明了$S_k^+(G)\leq e(G)+\binom{k+1}{2}$对$k=1, 2, \ldots, n.$的猜想。本文给出了$S_k^+(G),$的各种上界，并证明了这个猜想对下列情形成立：具有足够大$k,$的连通图，当$k\neq 3.$
$q$-Laplacian State Transfer on Graphs with Involutions
本文研究带对合的图的$q$-拉普拉斯矩阵上的状态转移，证明完美对状态转移等价于子图中的顶点状态转移，并构造了具有完美对状态转移的无限树族和单圈图族。
On the sign patterns of the smallest signless Laplacian eigenvector
设$H$是连通二部图，其无符号拉普拉斯矩阵为$Q(H)$ .设$H$的二分划是$(S,T)$，$x$是$Q(H)$的最小特征值的特征向量.众所周知，$x$在$S$上为正且恒定，在$T$上为负且恒定。本文研究了由$S$或$T$引入的子图中添加边时$x$的符号模式的恢复性，并描述了$x$的符号模式持续存在的一些情况。
This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs having the maximum (or minimum) values of graph invariants such as (Laplacian, p-Laplacian, signless Laplacian) spectral radius, the first Dirichlet eigenvalue, the Wiener index, the Harary index, the number of subtrees and the chromatic number etc, in given sets with the same tree, unicyclic, graphic degree sequences. Moreover, some conjectures are included.
A mixed graph $M_{G}$ is the graph obtained from an unoriented simple graph $G$ by giving directions to some edges of $G$, where $G$ is often called the underlying graph of $M_{G}$. In this paper, we introduce two classes of incidence matrices of the second kind of $M_{G}$, and discuss the determinants of these two matrices for rootless mixed trees and unicyclic mixed graphs. Applying these results, we characterize the explicit expressions of various minors for Hermitian (quasi-)Laplacian matrix of the second kind of $M_{G}$. Moreover, we give two sufficient conditions that the absolute values…
On the multiplicity of 1 as a Laplacian eigenvalue of a graph
本文研究图的拉普拉斯特征值1的重数，通过约化图方法证明约化树和约化单圈图的重数上界，并完全刻画达到上界的图类。
Laplacian Immanantal Polynomials of a Bipartite Graph and Graph Shift Operation
本文将树上的图移位操作推广为广义图移位操作，据此将已知不等式推广至二部图，并利用顶点定向证明相关定理与极值问题。
The list r-hued coloring of trees and unicyclic graphs
本文证明了树和大多数单圈图的列表r-色调色数等于min{r,Δ(G)}+1，并给出了例外情况的上界。
In a graph G; a vertex (resp. an edge) metric generator is a set of vertices S such that any pair of vertices (resp. edges) from G is distinguished by at least one vertex from S: The cardinality of a smallest vertex (resp. edge) metric generator is the vertex (resp. edge) metric dimension of G: In [19] we determined the vertex (resp. edge) metric dimension of unicyclic graphs and that it takes its value from two consecutive integers. Therein, several cycle configurations were introduced and the vertex (resp. edge) metric dimension takes the greater of the two consecutive values only if any of …
Substructure Analysis and Cycle Enumeration Methods for Oriented Graphs Based on Parameterizing Hermitian Laplacian Matrices by Galois Conjugates
本文研究有向图参数化Hermitian Laplacian矩阵的主子式，证明当参数取本原p次单位根的Galois共轭时行列式之积为p的整数幂，并据此提出枚举子结构中非消失单圈图分量的方法。
The Laplacian spread of a graph is the difference between the largest eigenvalue and the second-smallest eigenvalue of the Laplacian matrix of the graph. We find that the class of strongly regular graphs attains the maximum of largest eigenvalues, the minimum of second-smallest eigenvalues of Laplacian matrices and hence the maximum of Laplacian spreads among all simple connected graphs of fixed order, minimum degree, maximum degree, minimum size of common neighbors of two adjacent vertices and minimum size of common neighbors of two nonadjacent vertices. Some other extremal graphs are also pr…
The Laplacian matrix of a graph $G$ is denoted by $L(G)=D(G)-A(G)$, where $D(G)=diag(d(v_{1}),\ldots , d(v_{n}))$ is a diagonal matrix and $A(G)$ is the adjacency matrix of $G$. Let $G_1$ and $G_2$ be two graphs. A one-edge connection of two graphs $G_1$ and $G_2$ is a graph $G=G_1\odot_{uv} G_2$ with $V(G)=V(G_1)\cup V(G_2)$ and $E(G)= E(G_1)\cup E(G_2)\cup \{e=uv\}$, where $u\in V(G_1)$ and $v\in V(G_2)$. We investigate the multiplicity of the Laplacian eigenvalue $2$ of $G_1\odot_{uv} G_2$, while the unicyclic graphs $G_1$ and $G_2$ have $2$ among their Laplacian eigenvalues, by using their…
Graphs with zero as a main eigenvalue of the signless Laplacian
研究刻画恰好有\(\ell\geq3\)个\(Q\)-主特征值且其一为零的图，给出\(\ell = 3\)时的情况，还构造了无穷多个圈秩为\(k\)且直径无界的图，为相关图的分类提供反例。
Upper bound for the Laplacian eigenvalues of a graph
本文给出了无权图的Laplacian特征值的一个新的上界。设$G$是以$n$为顶点的单图.令$d_{m}(G)$和$λ_{m+1}(G)$分别为$G$的第$m$个最小度和$G$的第$m+1$个最小拉普拉斯特征值。然后为$\bar{G} \neq K_{m}+(n-m)K_1 $选择$ λ_{m+1}(G)\leq d_{m}(G)+m-1 $。我们还引入了赋权图的Laplacian特征值的上、下界，并将其与特殊情况的非赋权图进行了比较。
Let $q_{\min}(G)$ stand for the smallest eigenvalue of the signless Laplacian of a graph $G$ of order $n.$ This paper gives some results on the following extremal problem: How large can $q_\min\left( G\right) $ be if $G$ is a graph of order $n,$ with no complete subgraph of order $r+1?$ It is shown that this problem is related to the well-known topic of making graphs bipartite. Using known classical results, several bounds on $q_{\min}$ are obtained, thus extending previous work of Brandt for regular graphs. In addition, using graph blowups, a general asymptotic result about the maximum $q_{\m…
Let $T$ be a tree on $n$ vertices with $q$-Laplacian $L_{T}^{q}$. Let $GTS_n$ be the generalized tree shift poset on the set of unlabelled trees with $n$ vertices. We prove that for all $q \in R$, going up on $GTS_n$ has the following effect: the spectral radius and the second smallest eigenvalue of $L_{T}^{q}$ increase while the smallest eigenvalue of $L_{T}^{q}$ decreases. These generalize known results for eigenvalues of the Laplacian. As a corollary, we obtain consequences about the eigenvalues of $q,t$-Laplacians and exponential distance matrices of trees.
Let $A(G)$ and $D(G)$ be the adjacency matrix and the degree diagonal matrix of a graph $G$, respectively. Then $L(G)=D(G)-A(G)$ is called Laplacian matrix of the graph $G$. Let $G$ be a graph with $n$ vertices and $m$ edges. Then the $LI$-matrix of $G$ are defined as $LI(G)=L(G)-\frac{2m}{n}I_n$, where $I_n$ is the identity matrix. In this paper, we are interested in extremal properties of the Ky Fan $k$-norm of the $LI$-matrix of graphs, which is closely related to the well known problems and results in spectral graph theory, such as the Laplacian spectral radius, the Laplacian spread, the s…
Bipartite and neighborhood graphs and the spectrum of the normalized graph Laplacian
研究了连通图$Γ$的正规化Laplace算子的谱.众所周知，最小的非平凡特征值测量将$Γ$分解成两大块的难度，而最大的特征值控制$Γ$与二分的接近程度。最小的特征值可以控制的Cheeger常数，我们建立了一个对偶结构，控制最大的特征值。此外，我们发现的顺序$l\geq2$的邻域图$Γ[l]$编码重要的频谱信息$Γ$本身，我们系统地探讨。特别是，邻域图方法导致新的估计，可以改善Cheeger不等式的最小的非平凡特征值，以及明确估计的最大特征值从上面和下面。作为这种谱估计的应用，我们给出了耦合映象格子的同步性的一个判据，以及图上随机游动的收敛速度的一个估计。
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。
We consider the signless $p$-Laplacian of a graph, a generalisation of the usual signless Laplacian (the case $p=2$). We show a Perron-Frobenius property and basic inequalites for the largest eigenvalue and provide upper and lower bounds for the smallest eigenvalue in terms of a parameter related to the bipartiteness. The latter result generalises bounds by Desai and Rao and, interestingly, in the limit $p\to 1$ upper and lower bounds coincide.
A linear algebraic proof of the Laplacian spread conjecture
研究拉普拉斯谱展猜想，即\(\alpha(G)+\alpha(\overline{G}) \geq 1\)，核心方法是线性代数，主要贡献是给出更简洁的新证明。
Let $G$ be a simple connected undirected graph. The Laplacian spectral ratio of $G$, denoted by $R_L(G)$, is defined as the quotient between the largest and second smallest Laplacian eigenvalues of $G$, which is closely related to the structural parameters of a graph (or network), such as diameter, $t$-tough, perfect matching, average density of cuts, and synchronizability, etc. In this paper, we obtain some bounds of the Laplacian spectral ratio, which improves the known results. In addition, we give counter-examples on the upper bound of the Laplacian spectral ratio conjecture of trees, and …
Inverse Fiedler vector problem of a graph
本文研究图结构如何控制不同加权拉普拉斯矩阵的Fiedler向量，刻画了树和圈中所有可能的Fiedler向量及相关特征向量，并探讨了特征集在树上的分布。
Characterization of tricyclic graphs with exactly two $Q$-main eigenvalues
图$G$的无符号拉普拉斯矩阵定义为它的邻接矩阵与度对角矩阵之和，它的特征值称为图$G$的$Q$ -特征值.一个图$G$的$Q$ -特征值称为$Q$ -主特征值，如果它有一个特征向量的元素之和不等于零. Chen和Huang [L. Chen，Q.X. Huang，Trees，unicyclic graphs and bicyclic graphs with exactly two $Q$ -main eigenvalues，submitted for publication]分别刻画了所有树，单圈图和双圈图的两个$Q$-main eigenvalues.作为它的继续，本文刻划了具有恰好两个$Q$ -主特征值的所有三圈图.
For a graph $G$, the unraveled ball of radius $r$ centered at a vertex $v$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the $s$-th (where $s\ge 2$) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when $s=2$, the result may be regarded as an Alon--Boppana type bound for a class of irregular graphs.
The independence number, coloring number and related parameters are investigated in the setting of oriented hypergraphs using the spectrum of the normalized Laplace operator. For the independence number, both an inertia--like bound and a ratio--like bound are shown. A Sandwich Theorem involving the clique number, the vector chromatic number and the coloring number is proved, as well as a lower bound for the vector chromatic number in terms of the smallest and the largest eigenvalue of the normalized Laplacian. In addition, spectral partition numbers are studied in relation to the coloring numb…
The domination number and the least $Q$-eigenvalue
图$G$的顶点集$D$称为控制集，如果$V(G)\setminus D$的每个顶点都与$D$中的至少一个顶点相邻，控制数$γ(G)$（简称$γ$）是$G$的所有控制集的最小基数.图的最小$Q$ -特征值是它的无符号拉普拉斯矩阵的最小特征值。本文对一个既有阶数又有控制数的非二部图证明了$n$，并证明了它包含一个具有相同控制数$γ$的单圈生成子图.通过研究图的控制数与最小$Q$ -特征值之间的关系，我们在所有具有给定控制数的非二部图中极小化最小$Q$ -特征值.
Interpreting the von Neumann entropy of graph Laplacians, and coentropic graphs
对于任意图，我们在二分张量积空间上定义一个秩为1的算子，其分量分别与顶点集和边集相关联。我们证明了该算子的部分迹是拉普拉斯算子和边拉普拉斯算子。这提供了一个解释的冯诺依曼熵的（规范化）\拉普拉斯作为量子纠缠的数量在两个系统之间对应的顶点和边缘。在这个框架下，共谱图正好对应于局部酉等价纯态。最后，我们引入了共熵图的概念，即具有相等冯诺依曼熵的图。我们能够构造的最小的共熵（但不是共谱）图有8个顶点。具有n个顶点和m条边的共熵图的等价类数是具有相应维数的子系统的（纯）二部纠缠类数的下界。
Asymptotic behavior of the number of Eulerian orientations of graphs
我们考虑一类具有大代数连通度（拉普拉斯矩阵的第二小特征值）的单图。对于这类图，我们确定的欧拉方向的数量的渐近行为。此外，我们还建立了Laplacian矩阵的一些新的性质，以及渐近对角优势矩阵的条件性的一个估计
In this paper, we characterize all graphs with eigenvectors of the signless Laplacian and adjacency matrices with components equal to $\{- 1, 0, 1\}.$ We extend the graph parameter max $k$-cut to square matrices and prove a general sharp upper bound, which implies upper bounds on the max $k$-cut of a graph using the smallest signless Laplacian eigenvalue, the smallest adjacency eigenvalue, and the largest Laplacian eigenvalue of the graph. In addition, we construct infinite families of extremal graphs for the obtained upper bounds.
Nonpositive Eigenvalues of the Adjacency Matrix and Lower Bounds for Laplacian Eigenvalues
设$NPO(k)$是使任何顶点数为$n$或更多的无向图的邻接矩阵至少有$k$非正特征值的最小数$n$。我们证明了$NPO(k)$是定义良好的，并证明了$NPO(k)$对于$k=1,2,3,4,5$的值分别是$1,3,6,10,16$。另外，我们证明了对所有的$k \geq 5$，$R(k,k+1) \ge NPO(k) > T_k$，其中$R(k,k+1)$是$k$和$k+1$的Ramsey数，$T_k$是$k^{th}$三角数.这意味着Laplacian矩阵特征值的新的下界：$k$ -第一大特征值由$NPO(k)$ -第一大度从下到上有界，推广了一些先前的结果.
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