图的规范拉普拉斯谱与优超（英文）

令G是简单图.记L(G)为图G的规范拉普拉斯矩阵,其特征值称为图的规范拉普拉斯特征值.[Adv.Math.(China),2017,46(6):848-856]给出了关于规范拉普拉斯特征值和的相关结论,并提出相关猜想.我们发现在上述文章中的一些重要结果中存在一些错误.本文修正了所有不正确的结果.此外,我们讨论了￡(G)的特征值优超不等式.利用这些结果,我们证实了[Adv.Math.(China),2017,46(6):848-8561中提出的一个猜想.     

补图是树的图的距离拉普拉斯特征值（英文）

一个连通图的距离拉普拉斯矩阵定义为顶点传输度对角矩阵与距离矩阵的差,距离拉普拉斯矩阵的特征值称为这个图的距离拉普拉斯特征值.距离拉普拉斯伸展度定义为图的最大与次小距离拉普拉斯特征值的差.本文确定了补图的最大距离拉普拉斯特征值取得最小值和最大值的树及补图的次小距离拉普拉斯特征值取得最小值和最大值的树,也确定了补图的次大距离拉普拉斯特征值取得最小值的树,还确定了补图的距离拉普拉斯伸展度取得最小值和最大值的树.     

可迹图的谱充分条件

设G是一个简单图,A(G),Q(G)以及Q(G)分别为G的邻接矩阵,无符号拉普拉斯矩阵以及距离无符号拉普拉斯矩阵,其最大特征值分别称为G的谱半径,无符号拉普拉斯谱半径以及距离无符号拉普拉斯谱半径.如果图G中有一条包含G中所有顶点的路,则称这条路为哈密顿路;如果图G含有哈密顿路,则称G为可迹图;如果图G含有从任意一点出发的哈密顿路,则称G从任意一点出发都是可迹的.主要研究利用图G的谱半径,无符号拉普拉斯谱半径,以及距离无符号拉普拉斯谱半径,分别给出图G从任意一点出发都是可迹的充分条件.     

剖分点—边冠图的谱(英文)

在三个图G_1,G_2和G_3的基础上引进了一种新的图运算,在这种运算下得到的图称为剖分点一边冠图,记作G_1~S·(G_2~V∪G_3~E).同时,当G_1是正则图时,确定了剖分点一边冠图的邻接谱和拉普拉斯谱.作为应用,构造了无穷多对邻接同谱图和拉普拉斯同谱图.此外,还确定了剖分点一边冠图的生成树的个数.     

基于拉普拉斯谱确定的两类树

设图G是简单连通图.如果任何一个与图G关于拉普拉斯矩阵同谱的图,都与图G同构,称图G可由其拉普拉斯谱确定.定义了树Y_n和树F(2,n,1)两类特殊结构的树.利用同谱图线图的特点,证明了树Y_n和树F(2,n,1)可由其拉普拉斯谱确定.     

关于图的拟拉普拉斯谱半径

用代数方法给出了一个关于简单图的顶点度数与拟拉普拉斯谱半径的不等式,并给出了图的拟拉普拉斯谱半径的一个新上界.     

一些极值图问题的谱条件综述（英文）

这篇综述分为两个方面.首先,我们总结了图论中的Turan型问题的谱极值结论的最新进展.更准确地说,关于各种图的邻接谱半径和无符号拉普拉斯谱半径,我们总结了它们的谱版本的Turán型函数.例如,完全图、色数至少为3的一般图、完全二部图、奇圈、偶圈、色临界图和相交三角形图.第二个目标是总结一些最近的关于图性质的谱条件.通过一种统一的方法,基于邻接谱半径和无符号拉普拉斯谱半径,我们给出了一些充分条件,使得该图成为哈密顿图、k-哈密顿图、k-边哈密顿图、可迹图、k-路径可覆盖图、k-连通图、k-边连通图、哈密顿连通图、完美匹配图和β-亏量图.     

马氏调制散粒噪声模型在保险中的应用

在常利率环境下,研究了一个含相关业务类的复合Cox风险模型,其保险业务间的相关结构是由共同跳结构来描述的.假设两类保险业务的理赔来到强度过程服从一个多维的马氏调控散粒噪声过程.利用鞅方法,给出了总折现理赔量和马氏调控散粒噪声强度过程的联合拉普拉斯变换,并利用拉普拉斯变换给出了总折现理赔量的期望和方差等特征量.     

具有固定直径的树的拉普拉斯谱半径

一个图称为毛毛虫,如果从它删去所有的悬挂点后得到的图是一个路.研究了具有固定直径的毛毛虫树的拉普拉斯谱半径,确定了其中具有最大拉普拉斯谱半径的毛毛虫树并且讨论了该树的一些性质.     

最近点图的密度矩阵可分的充分必要条件

图的密度矩阵是迹为1的组合拉普拉斯矩阵.本文通过对图的密度矩阵组合条件("度数条件")的研究来判断其可分性,从立体几何的角度直观的证明了该条件为n=mpg个顶点上的图的密度矩阵三体可分的一个充分必要条件.     

On the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph

The Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph are the characteristic polynomials of its Laplacian matrix, signless Laplacian matrix and normalized Laplacian matrix, respectively. In this paper, we mainly derive six reduction procedures on the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph which can be used to construct larger Laplacian, signless Laplacian and normalized Laplacian cospectral graphs, respectively.     

Normalized Laplacian spectrum of some subdivision-coronas of two regular graphs

In this paper, we determine the full normalized Laplacian spectrum of the subdivision-vertex corona, subdivision-edge corona, subdivision-vertex neighbourhood corona and subdivision-edge neighbourhood corona of a connected regular graph with an arbitrary regular graph in terms of their normalized Laplacian eigenvalues. Moreover, applying these results, we find some non-regular normalized Laplacian cospectral graphs.     

A note on normalized Laplacian energy of graphs

Themain goal of this paper is to obtain some bounds for the normalized Laplacian energy of a connected graph. The normalized Laplacian energy of the line and para-line graphs of a graph are investigated. The relationship of the smallest and largest positive normalized Laplacian eigenvalues of graphs are also studied.     

Effects on the normalized Laplacian spectral radius of non-bipartite graphs under perturbation and their applications

The normalized Laplacian eigenvalues of a network play an important role in its structural and dynamical aspects associated with the network. In this paper, we consider how the normalized Laplacian spectral radius of a non-bipartite graph behaves by several graph operations. As an example of the application, the smallest normalized Laplacian spectral radius of non-bipartite unicyclic graphs with fixed order is determined.     

The Laplacian incidence energy of graphs

The Laplacian incidence energy of a graph is defined as the sum of the singular values of its normalized oriented incidence matrix. In this paper, we give sharp upper and lower bounds as well as the Coulson integral formula for the Laplacian incidence energy. Moreover, we show a close relation of the Laplacian incidence energy, normalized incidence energy and Randi? energy.     

Spectral clustering and its use in bioinformatics

We formulate a discrete optimization problem that leads to a simple and informative derivation of a widely used class of spectral clustering algorithms. Regarding the algorithms as attempting to bi-partition a weighted graph with N vertices, our derivation indicates that they are inherently tuned to tolerate all partitions into two non-empty sets, independently of the cardinality of the two sets. This approach also helps to explain the difference in behaviour observed between methods based on the unnormalized and normalized graph Laplacian. We also give a direct explanation of why Laplacian eigenvectors beyond the Fiedler vector may contain fine-detail information of relevance to clustering. We show numerical results on synthetic data to support the analysis. Further, we provide examples where normalized and unnormalized spectral clustering is applied to microarray data—here the graph summarizes similarity of gene activity across different tissue samples, and accurate clustering of samples is a key task in bioinformatics.     

Resistance distance and the normalized Laplacian spectrum

It is well known that the resistance distance between two arbitrary vertices in an electrical network can be obtained in terms of the eigenvalues and eigenvectors of the combinatorial Laplacian matrix associated with the network. By studying this matrix, people have proved many properties of resistance distances. But in recent years, the other kind of matrix, named the normalized Laplacian, which is consistent with the matrix in spectral geometry and random walks [Chung, F.R.K., Spectral Graph Theory, American Mathematical Society: Providence, RI, 1997], has engendered people's attention. For many people think the quantities based on this matrix may more faithfully reflect the structure and properties of a graph. In this paper, we not only show the resistance distance can be naturally expressed in terms of the normalized Laplacian eigenvalues and eigenvectors of G, but also introduce a new index which is closely related to the spectrum of the normalized Laplacian. Finally we find a non-trivial relation between the well-known Kirchhoff index and the new index.     

Eigenvalues of 2-edge-coverings

A 2-edge-covering between G and H is an onto homomorphism from the vertices of G to the vertices of H so that each edge is covered twice and edges in H can be lifted back to edges in G. In this note we show how to compute the spectrum of G by computing the spectrum of two smaller graphs, namely a (modified) form of the covered graph H and another graph which we term the anti-cover. This is done for both the adjacency matrix and the normalized Laplacian. We also give an example of two anti-cover graphs which have the same normalized Laplacian, and state a generalization for directed graphs.     

Graph spectra as a systematic tool in computational biology

We present the spectrum of the (normalized) graph Laplacian as a systematic tool for the investigation of networks, and we describe basic properties of eigenvalues and eigenfunctions. Processes of graph formation like motif joining or duplication leave characteristic traces in the spectrum. This can suggest hypotheses about the evolution of a graph representing biological data.     

A note about cospectral graphs for the adjacency and normalized Laplacian matrices

In this note we show how to construct two distinct bipartite graphs which are cospectral for both the adjacency and normalized Laplacian matrices by ‘unfolding’ a base bipartite graph in two different ways.