D. M. Cardoso, M. A. A. de Freitas, E. A. Martins, M. Robbinao, B. San Martín
The characteristic polynomials of the adjacency matrix of line graphs of caterpillars and then the characteristic polynomials of their Laplacian or signless Laplacian matrices are characterized, using recursive formulas. Furthermore, the obtained results are applied on the determination of upper and lower bounds on the algebraic connectivity of these graphs.
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We consider certain matrix-products where successive matrices in the product belong alternately to a particular qualitative class or its transpose. The main theorems relate structural and spectral properties of these matrix-products to the structure of underlying bipartite graphs. One consequence is a characterisation of caterpillars: a graph is a caterpillar if and only if all matrix-products associated with it have real nonnegative spectrum. Several other equivalences of this kind are proved. The work is inspired by certain questions in dynamical systems where such products arise naturally a…
Suppose that the vertex set of a connected graph $G$ is $V(G)=\{v_1,\cdots,v_n\}$. Then we denote by $Tr_{G}(v_i)$ the sum of distances between $v_i$ and all other vertices of $G$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry equal to $Tr_{G}(v_{i})$ and $D(G)$ be the distance matrix of $G$. Then $Q_{D}(G)=Tr(G)+D(G)$ is the distance signless Laplacian matrix of $G$. The largest eigenvalues of $Q_D(G)$ is called distance signless Laplacian spectral radius of $G$. In this paper we give some graft transformations on distance signless Laplacian spectral radius of the grap…
Limit points of (signless) Laplacian spectral radii of linear trees
本研究将Shearer的毛虫树邻接极限点方法适配用于图拉普拉斯谱半径极限点研究，发现原方法存在无极限点的区间，进而推广至线性树拉普拉斯极限点得到更大极限点集，为相关稠密性证明提供工具。
Distribution of deformed Laplacian limit points
本文研究了变形拉普拉斯算子的极限点分布，证明了对于特定参数s，区间[λ_0, +∞)完全由s-变形拉普拉斯算子的极限点构成。
In this paper, we obtain the bounds of the extreme eigenvalues of a normalized and signless Laplacian matrices using by their traces. In addition, we determine the bounds for k-th eigenvalues of normalized and signless Laplacian matrices.
On the Eccentricity Laplacian and Eccentricity Signless Laplacian Matrices of a Graph
本文引入了图的偏心度矩阵的拉普拉斯和无符号拉普拉斯矩阵，探讨了不同图类中偏心度拉普拉斯、偏心度无符号拉普拉斯和偏心度谱之间的等价性，并通过偏心度谱的对称性和这些拉普拉斯矩阵的相似性对E-双图进行谱刻画。
In [Steve Butler. A note about cospectral graphs for the adjacency and normalized Laplacian matrices. Linear Multilinear Algebra, 58(3-4):387-390, 2010.], Butler constructed a family of bipartite graphs, which are cospectral for both the adjacency and the normalized Laplacian matrices. In this article, we extend this construction for generating larger classes of bipartite graphs, which are cospectral for both the adjacency and the normalized Laplacian matrices. Also, we provide a couple of constructions of non-bipartite graphs, which are cospectral for the adjacency matrices but not necessaril…
A signed graph is a graph whose edges are labeled either positive or negative. Corresponding to the two signed distance matrices defined for signed graphs, we define two signed distance laplacian matrices. We characterize balance in signed graphs using these matrices and find signed distance laplacian spectra of some classes of unbalanced signed graphs.
We show that all off-diagonal entries in the Moore-Penrose inverse of the distance Laplacian matrix of a tree are non-positive.
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.
The trees, respectively unicyclic graphs, on $n$ vertices with the smallest Laplacian permanent are studied. In this paper, by edge-grafting transformations, the $n$-vertex trees of given bipartition having the second and third smallest Laplacian permanent are identified. Similarly, the $n$-vertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Laplacian permanent are characterized. Consequently, the $n$-vertex bipartite unicyclic graphs with the first, second and third smallest Laplacian permanent are determined.
Forest matrices around the Laplacian matrix
本文研究了有向图G的森林内矩阵Q_k及其与G的拉普拉斯矩阵L的关系。Q_k的（i，j）项是具有k条弧的生成收敛森林（in-forests）的总权重，使得i属于根在j的树。森林矩阵Q_k可以递归计算并用拉普拉斯矩阵中的多项式表示;它们提供了L的广义逆、幂和一些特征向量的表示。正规化的林内矩阵是行随机的，最大林内矩阵的正规化矩阵是Laplacian矩阵的特征投影，这为马尔可夫链树定理提供了一个直接的证明.这些结果的一个来源是矩阵Q_k是adj（a*I+L）的多项式展开式中的矩阵系数。因此，它们是-L的Faddeev矩阵。保留字：加权有向图;拉普拉斯矩阵;跨越森林;矩阵森林定理; Leverrier-Faddeev方法;马尔可夫链树定理;特征投影;广义逆;奇异M-矩阵
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。
Let $G=(V,\overrightarrow{E})$ be a graph with some prescribed orientation for the edges and $Γ$ be an arbitrary group. If $f\in \mathrm{Inv}(Γ)$ be an anti-involution then the skew gain graph $Φ_f=(G,Γ,φ,f)$ is such that the skew gain function $φ:\overrightarrow{E}\rightarrow Γ$ satisfies $φ(\overrightarrow{vu})=f(φ(\overrightarrow{uv}))$. In this paper, we study two different types, Laplacian and $g$-Laplacian matrices for a skew gain graph where the skew gains are taken from the multiplicative group $F^\times$ of a field $F$ of characteristic zero. Defining incidence matrix, we also prove t…
We provide upper and lower bounds on the smallest eigenvalue of grounded Laplacian matrices (which are matrices obtained by removing certain rows and columns of the Laplacian matrix of a given graph). The gap between the upper and lower bounds depends on the ratio of the smallest and largest components of the eigenvector corresponding to the smallest eigenvalue of the grounded Laplacian. We provide a graph-theoretic bound on this ratio, and subsequently obtain a tight characterization of the smallest eigenvalue for certain classes of graphs. Specifically, for Erdos-Renyi random graphs, we show…
Inverse eigenvalue problem for Laplacian matrices of a graph
该研究针对图的广义拉普拉斯矩阵逆特征值问题，探究特定图族与少顶点图的可实现谱、有序重数列表，并从理论与数值层面分析规模归一化加权下该矩阵族的最小方差。
The \emph{distance matrix} of a simple connected graph $G$ is $D(G)=(d_{ij})$, where $d_{ij}$ is the distance between the vertices $i$ and $j$ in $G$. We consider a weighted tree $T$ on $n$ vertices with edge weights are square matrix of same size. The distance $d_{ij}$ between the vertices $i$ and $j$ is the sum of the weight matrices of the edges in the unique path from $i$ to $j$. In this article we establish a characterization for the trees in terms of rank of (matrix) weighted Laplacian matrix associated with it. Then we establish a necessary and sufficient condition for the distance matr…
The Smith normal form of Laplacian matrices of simplicial annuli and high dimensional trees
研究单纯复形拉普拉斯矩阵的史密斯标准型，通过建立邻接图沙堆群与该标准型的关系，计算单纯环面和\(k -\)树的相关标准型，经数值实验可视化其区分\(k -\)树的效果，并指出相关矩阵在拉普拉斯矩阵中的应用。
Spectral properties of the deformed Laplacian matrix of trees and H-join graphs
本文研究了树和H-连接图的变形拉普拉斯矩阵的谱性质，推导了树的特征值定位结果，并计算了H-连接图的谱。
The Strong Spectral Property and the Jacobian Method for Weighted Laplacian Matrices
本文提出加权拉普拉斯矩阵的强谱属性，结合雅可比方法分析谱区域边界，并探讨其在图论中的应用。
Graph Laplacians with Higher Accuracy
本文受数值分析中高阶精度离散拉普拉斯算子启发，提出一类新的高精度图拉普拉斯矩阵，证明其能显著减少同谱伴侣图的数量，并研究其谱性质及带权符号图的存在性。
Two Laplacians for the resistance distance matrix of a graph
针对连通图的电阻距离矩阵，本文定义电阻拉普拉斯矩阵与电阻无符号拉普拉斯矩阵，分析其在典型图类中的形式、谱性质，并提出电阻拉普拉斯能量及其界。
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。
Graphs can be associated with a matrix according to some rule and we can find the spectrum of a graph with respect to that matrix. Two graphs are cospectral if they have the same spectrum. Constructions of cospectral graphs help us establish patterns about structural information not preserved by the spectrum. We generalize a construction for cospectral graphs previously given for the distance Laplacian matrix to a larger family of graphs. In addition, we show that with appropriate assumptions this generalized construction extends to the adjacency matrix, combinatorial Laplacian matrix, signles…
M. Aouchiche and P. Hansen proposed the distance Laplacian and the distance signless Laplacian of a connected graph [Two Laplacians for the distance matrix of a graph, LAA 439 (2013) 21{33]. In this paper, we obtain three theorems on the sharp upper bounds of the spectral radius of a nonnegative matrix, then apply these theorems to signless Laplacian matrices and the distance signless Laplacian matrices to obtain some sharp bounds on the spectral radius, respectively. We also proposed a known result about the sharp bound of the signless Laplacian spectral radius has a defect.
Generalized block diagonal Laplacian spectrum of graphs
本文研究了图的广义块对角拉普拉斯谱，将$p^2$块全一矩阵简化为$p$块全一矩阵，并探讨了有向图的海森堡邻接矩阵的类比
Doubly Stochastic Matrices and Modified Laplacian Matrices of Graphs
本文研究图拉普拉斯矩阵加单位矩阵后所得正定矩阵的逆（双随机矩阵），建立其元素单调性、路径图与Fibonacci数的关系、树的对角下界及高效算法，并揭示其与热方程等偏微分方程的联系。
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。
A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the distance matrix, distance Laplacian, and distance signless Laplacian, in addition to the adjacency matrix, Laplacian, and signless Laplacian. Various sums of Kronecker products of nonnegative matrices are introduced to model the Cartesian and lexicographic products of digraphs. The Jordan canonical form is applied extensively to the analysis of spectra and eigenvectors. The analysis shows that Cartesian products provide a method…
In this paper, we give the spectrum of a matrix by using the quotient matrix, then we apply this result to various matrices associated to a graph and a digraph, including adjacency matrix, (signless) Laplacian matrix, distance matrix, distance (signless) Laplacian matrix, to obtain some known and new results. Moreover, we propose some problems for further research.
On the spectra of nonsymmetric Laplacian matrices
拉普拉斯矩阵是具有非正非对角元素和零行和的正方形实矩阵。作为与加权有向图相联系的矩阵，它推广了普通图的拉普拉斯矩阵。标准化拉普拉斯矩阵是非对角项的绝对值不超过1/n的拉普拉斯矩阵，其中n是矩阵的阶。研究了Laplacian矩阵的谱以及Laplacian矩阵与随机阵的关系。本文证明了标准化Laplacian矩阵是收敛的。标准化拉普拉斯矩阵的特征值0和1的重数分别等于相应有向图的森林维数和互补有向图的森林维数之小于。这些特征值是半单的。标准化拉普拉斯矩阵的谱属于两个闭合圆盘的交点，一个以1/n为中心，另一个以1-1/n为中心，每个圆盘的半径为1- 1/n，还有两个闭合角，一个以从1画出的两条半直线为界，另一个以从0画出的两条半直线为界，通过某些点。本征值的虚部由1/（2n）cot（pi/2n）限定;当n趋于无穷大时，这个最大值收敛到1/pi。关键词：Laplacian矩阵;图的Laplacian图谱;加权有向图;有向图的森林维数;随机阵
In this paper, we introduce a generalization of corona of graphs. This construction generalizes the generalized corona of graphs (consequently, the corona of graphs), the cluster of graphs, the corona-vertex subdivision graph of graphs and the corona-edge subdivision graph of graphs. Further, it enables to get some more variants of corona of graphs as its particular cases. To determine the spectra of the adjacency, Laplacian and the signless Laplacian matrices of the above mentioned graphs, we define a notion namely, the coronal of a matrix constrained by an index set, which generalizes the co…
Let $f(D(i, j), d_i, d_j)$ be a real function symmetric in $i$ and $j$ with the property that $f(d, (1+o(1))np, (1+o(1))np)=(1+o(1))f(d, np, np)$ for $d=1,2$. Let $G$ be a graph, $d_i$ denote the degree of a vertex $i$ of $G$ and $D(i, j)$ denote the distance between vertices $i$ and $j$ in $G$. In this paper, we define the $f$-weighted Laplacian matrix for random graphs in the Erd$\ddot{o}$s-R$\acute{e}$nyi random graph model $\mathcal{G}_{n, p}$, where $p\in (0, 1)$ is fixed. Four weighted Laplacian type energies: the weighted Laplacian energy $\mathscr{LE}_f(G)$, weighted signless Laplacian…
Expansion in Distance Matrices
本文研究归一化距离拉普拉斯矩阵，证明其谱隙与 Cheeger 常数均独立于具体图而远离零，并将谱结论推广到有限度量空间。
We give upper and lower bounds for the spectral radius of a nonnegative matrix by using its average 2-row sums, and characterize the equality cases if the matrix is irreducible. We also apply these bounds to various nonnegative matrices associated with a graph, including the adjacency matrix, the signless Laplacian matrix, the distance matrix, the distance signless Laplacian matrix, and the reciprocal distance matrix.
On digraphs determined by their singular values
本文研究了有向路径、有向环和星形图的奇异值，证明了有向路径、有向环和定向星形图由拉普拉斯和无符号拉普拉斯奇异值确定，但不通过邻接矩阵奇异值。
New matrices for spectral hypergraph theory, II
本文引入三种基于统一矩阵的超图矩阵，证明其与关联图对应矩阵等同，从而通过矩阵谱建立超图与关联图结构性质的联系，并引入相关超图结构与不变量。
The $Z_q$-forcing number for some graph families
本文引入了带惯性限制的$Z_q$-强迫游戏，证明了树的上界，并利用博弈论方法给出了$n$顶点毛毛虫图$Z_q$-强迫数的上下界证明。
We discuss the connections tying Laplacian matrices to abstract duality and planarity of graphs.
We present new short proofs of known spanning tree enumeration formulae for threshold and Ferrers graphs by showing that the Laplacian matrices of such graphs admit triangular rank-one perturbations. We then characterize the set of graphs whose Laplacian matrices admit triangular rank-one perturbations as the class of special 2-threshold graphs, introduced by Hung, Kloks, and Villaamil. Our work introduces (1) a new characterization of special 2-threshold graphs that generalizes the characterization of threshold graphs in terms of isolated and dominating vertices, and (2) a spanning tree enume…
We study the sets of inertias achieved by Laplacian matrices of weighted signed graphs. First we characterize signed graphs with a unique Laplacian inertia. Then we show that there is a sufficiently small perturbation of the nonzero weights on the edges of any connected weighted signed graph so that all eigenvalues of its Laplacian matrix are simple. Next, we give upper bounds on the number of possible Laplacian inertias for signed graphs with a fixed flexibility $τ$ (a combinatorial parameter of signed graphs), and show that these bounds are sharp for an infinite family of signed graphs. Fina…
We present an elementary proof of a generalization of Kirchoff's matrix tree theorem to directed, weighted graphs. The proof is based on a specific factorization of the Laplacian matrices associated to the graphs, which only involves the two incidence matrices that capture the graph's topology. We also point out how this result can be used to calculate principal eigenvectors of the Laplacian matrices.
Unifying adjacency, Laplacian, and signless Laplacian theories
本文通过引入 $B_\alpha(G)=\alpha A(G)+(1-\alpha)L(G)$ 矩阵族，统一研究图的邻接、拉普拉斯与无符号拉普拉斯谱理论，并利用特征值关于 $\alpha$ 的连续性得到独立数上界、色数界及特征多项式的 Sachs 型结果。
Signed graphs and inverses of their incidence matrices
该研究针对符号图拉普拉斯矩阵的可逆性问题，通过推导符号图关联矩阵的Moore-Penrose逆的组合公式，得到拉普拉斯矩阵Moore-Penrose逆的组合公式，推广了无符号图的相关已知结果，并给出实例验证公式实用性。
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…
Coalescing sets preserving cospectrality of graphs arising from block similarity matrices
本文研究图的合并操作何时保持同谱性，通过相似矩阵的块结构给出充分条件，并覆盖邻接、拉普拉斯、无符号拉普拉斯、距离及广义距离矩阵等类型。
Limit points of $A_α$-matrices of graphs
本研究聚焦图的$A_α$-矩阵谱半径的极限点问题，改进J. B. Shearer的经典方法，证明了α接近0时毛毛虫树对应极限点的稠密性，明确了相关阈值及其他全为极限点的区间，丰富了图谱理论的相关结论。
It is known from the algebraic graph theory that if $L$ is the Laplacian matrix of some tree $G$ with a vertex degree sequence $\mathbf{d}=(d_1, ..., d_n)^\top$ and $D$ is its distance matrix, then $LD+2I=(2\cdot\mathbf{1}-\mathbf{d})\mathbf{1}^\top$, where $\mathbf{1}$ is an all-ones column vector. We prove that if this matrix identity holds for the Laplacian matrix of some graph $G$ with a degree sequence $\mathbf{d}$ and for some matrix $D$, then $G$ is essentially a tree, and $D$ is its distance matrix. This result immediately generalizes to weighted graphs. If the matrix $D$ is symmetric,…
Let $G$ be a simple graph. In 1986, Herbert Wilf asked what kind of graphs have an eigenvector with entries formed only by $\pm 1$? In this paper, we answer this question for the adjacency, Laplacian and signless Laplacian matrix of a graph. Besides, we generalize the concept of an exact graph to the adjacency and signless Laplacian matrices. Infinity families of exact graphs for all those matrices are presented.
Laplacian eigenvalues of independence complexes via additive compound matrices
本文通过加性复合矩阵建立独立复形拉普拉斯特征值与图拉普拉斯特征值的新下界，并由此给出约化同调群消失的充分条件，推广了已有结果。
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian matrices of an oriented hypergraph which depend on structural parameters of the oriented hypergraph are found. An oriented hypergraph and its incidence dual are shown to have the same nonzero Laplacian eigenvalues. A family of oriented hypergraphs with uniformally labeled incidences is also studied. This family provides a hypergraphic generalization of the s…
New matrices for the spectral theory of mixed graphs, Part II
本文在混合图集成邻接矩阵基础上，提出集成拉普拉斯矩阵、集成无符号拉普拉斯矩阵和归一化集成拉普拉斯矩阵，并研究其谱与混合图结构性质的关系。
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 determinants of $\{\pm 1\}$-matrices are calculated by via the oriented hypergraphic Laplacian and summing over an incidence generalization of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on their hyperedge containment. Every non-edge-monic family is shown to contribute a net value of $0$ to the Laplacian, while each edge-monic family is shown to sum to the absolute value of the determinant of the original incidence matrix. Simple symmetries are identified as well as their relationship to Hadamard's maximum determinant problem. Finally, the entries…
A matching M is a dominating induced matching of a graph, if every edge of the graph is either in $M$ or has a common end-vertex with exactly one edge in $M$. The concept of complete dominating induced matching is introduced as graphs where the vertex set can be partitioned into two subsets, one of them inducing an $1$-regular graph and the other defining an independent set and such that all the remaining edges connect each vertex of one set to each vertex of the other. The principal eigenvectors of the adjacency, Laplacian and signless Laplacian matrices of graphs with complete dominating ind…
We present enumeration results on the number of connected graphs up to 10 vertices for which there is at least one other graph with the same spectrum (a cospectral mate), or at least one other graph with the same Smith normal form (coinvariant mate) with respect to several matrices associated to a graph. The present data give some indication that possibly the Smith normal form of the distance Laplacian and the signless distance Laplacian matrices could be a finer invariant to distinguish graphs in cases where other algebraic invariants, such as those derived from the spectrum, fail. Finally, w…
For a given hypergraph, an orientation can be assigned to the vertex-edge incidences. This orientation is used to define the adjacency and Laplacian matrices. In addition to studying these matrices, several related structures are investigated including the incidence dual, the intersection graph (line graph), and the 2-section. A connection is then made between oriented hypergraphs and balanced incomplete block designs.
An oriented hypergraph is an oriented incidence structure that generalizes and unifies graph and hypergraph theoretic results by examining its locally signed graphic substructure. In this paper we obtain a combinatorial characterization of the coefficients of the characteristic polynomials of oriented hypergraphic Laplacian and adjacency matrices via a signed hypergraphic generalization of basic figures of graphs. Additionally, we provide bounds on the determinant and permanent of the Laplacian matrix, characterize the oriented hypergraphs in which the upper bound is sharp, and demonstrate tha…
I discuss the work of many authors on various matrices used to study signed graphs, concentrating on adjacency and incidence matrices and the closely related topics of Kirchhoff (`Laplacian') matrices, line graphs, and very strong regularity.
The laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states
研究了由组合拉普拉斯算子得到的混合密度矩阵的纠缠特性。这是通过引入图的密度矩阵的概念来实现的。我们刻画了具有纯密度矩阵的图，证明了图的密度矩阵总是可以写成图的纯密度矩阵的均匀混合。我们认为这些矩阵的冯诺伊曼熵，我们的最小值和最大值达到的图形的特征。然后我们讨论了图的可分离性问题，指出图的密度矩阵的可分离性并不总是依赖于顶点的标号。我们考虑图的张量积结构和简单的情况下，组合属性与纠缠的状态。我们计算了所有四个代表纠缠态的顶点上的图的并发度。事实证明，对于这些图中的一些图，并发度的值正好是分数。
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