On bipartite graphs with small number of laplacian eigenvalues greater than two and three

All connected bipartite graphs with exactly two Laplacian eigenvalues greater than two are determined. Besides, all connected bipartite graphs with exactly one Laplacian eigenvalue greater than three are determined.     

GRAPHS CHARACTERIZED BY LAPLACIAN EIGENVALUES

§1. Introduction Let G = (V,E) be a simple graph. The Laplacian matrix of G is L(G) = D(G)?A(G),where D(G) = diag (du,u ∈V (G)) (du is the degree of a vertex u) and A(G) are the degreediagonal and the adjacency matrices of G. The eigenvalues of L(G) are called the Laplacianeigenvalues and denoted by λ1(G) ≥λ2(G) ≥···≥λn(G) = 0or for short λ1 ≥λ2 ≥···≥λn = 0.The Laplacian matrix of a simple gra…     

Non-bipartite Graphs with Third Largest Laplacian Eigenvalue Less Than Three

All bipartite graphs whose third largest Laplacian eigenvalue is less than 3 have been characterized by Zhang. In this paper, all connected non-bipartite graphs with third largest Laplacian eigenvalue less than three are determined.     

Graphs with four distinct Laplacian eigenvalues

In this paper, we investigate connected nonregular graphs with four distinct Laplacian eigenvalues. We characterize all such graphs which are bipartite or have exactly one multiple Laplacian eigenvalue. Other examples of interest are also presented.     

给定染色数的无符号Laplace谱半径

设Gkn(k≥2)为n阶的染色数为k的连通图的集合.本文确定了Gkn中具有极大无符号Laplace谱半径的图,即k=2时为完全二部图,k≥3时为Turn图.本文也讨论了Gkn中的具有极小无符号Laplace谱半径的图,对k≤3的情形给出了此类图的刻画.     

含偶圈图的Laplacian 谱刻画

设 H(K_{1,5},P_n,C_l)是由路 P_n的两个悬挂点分别粘上星图K_{1,5}的悬挂点和圈 C_l的点所得的单圈图. 若两个二部图是关于Laplacian 矩阵同谱的, 则它们的线图是邻接同谱的, 两个邻接同谱图含有相同数目的同长闭回路. 如果任何一个与图G关于Laplacian 同谱图都与图G 同构, 那么称图G可由其Laplacian 谱确定. 利用图与线图之间的关系证明了H(K_{1,5},P_n,C_4)、H(K_{1,5},P_n,C_6) 由它们的Laplacian谱确定.     

双圈图的Laplace谱跨度

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In our recent work, we have determined the graphs with maximal Laplacian spreads among all trees of fixed order and among all unicyclic graphs of fixed order, respectively. In this paper, we continue the work on Laplacian spread of graphs, and prove that there exist exactly two bicyclic graphs with maximal Laplacian spread among all bicyclic graphs of fixed order, which are obtained from a star by adding two incident edges and by adding two nonincident edges between the pendant vertices of the star, respectively.     

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.     

Spectra of Corona Based on the Total Graph

For two simple connected graphs $G_1$ and $G_2$, we introduce a new graph operation called the total corona $G_1⊛G_2$ on $G_1$ and $G_2$ involving the total graph of $G_1.$ Subsequently, the adjacency (respectively, Laplacian and signless Laplacian) spectra of $G_1⊛G_2$ are determined in terms of these of a regular graph $G_1$ and an arbitrary graph $G_2.$ As applications, we construct infinitely many pairs of adjacency (respectively, Laplacian and signless Laplacian) cospectral graphs. Besides we also compute the number of spanning trees of $G_1⊛G_2.$     

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.     

拟树图按第二大无符号拉普拉斯特征值排序

A connected graph G=(V,E) is called a quasi-tree graph if there exists a vertex v_0∈V(G) such that G-v_0 is a tree.In this paper,we determine all quasi-tree graphs of order n with the second largest signless Laplacian eigenvalue greater than or equal to n-3.As an application,we determine all quasi-tree graphs of order n with the sum of the two largest signless Laplacian eigenvalues greater than to 2 n-5/4.     

The signless Laplacian spectral radius of bicyclic graphs with prescribed degree sequences

In this paper, we characterize all extremal connected bicyclic graphs with the largest signless Laplacian spectral radius in the set of all connected bicyclic graphs with prescribed degree sequences. Moreover, the signless Laplacian majorization theorem is proved to be true for connected bicyclic graphs. As corollaries, all extremal connected bicyclic graphs having the largest signless Laplacian spectral radius are obtained in the set of all connected bicyclic graphs of order n (resp. all connected bicyclic graphs with a specified number of pendant vertices, and all connected bicyclic graphs with given maximum degree).     

On the Laplacian spectral radii of bipartite graphs

The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we provide structural and behavioral details of graphs with maximum Laplacian spectral radius among all bipartite connected graphs of given order and size. Using these results, we provide a unified approach to determine the graphs with maximum Laplacian spectral radii among all trees, and all bipartite unicyclic, bicyclic, tricyclic and quasi-tree graphs, respectively.     

图的Laplacian特征值

本文研究了连通图的Laplacian特征值,利用图的Laplacian矩阵的特征多项式的行列式表示式,对存在两个不同顶点,但有相同邻集的一类图,得到了一个Laplacian特征值,并给出了它的应用.     

给定围长的三圈图的无符号拉普拉斯谱半径

A tricyclic graph G =（V（G）, E（G）） is a connected and simple graph such that｜E（G）｜ = ｜V（G）｜＋2. Let Tg nbe the set of all tricyclic graphs on n vertices with girth g. In this paper, we will show that there exists the unique graph which has the largest signless Laplacian spectral radius among all tricyclic graphs with girth g containing exactly three（resp., four）cycles. And at the same time, we also give an upper bound of the signless Laplacian spectral radius and the extremal graph having the largest signless Laplacian spectral radius in Tg n,where g is even.     

恰有两个Q-主特征值的三圈图

图G的无符号拉普拉斯矩阵定义为图G的邻接矩阵与度对角矩阵的和,其特征值称为图G的Q-特征值.图G的一个Q-特征值称为Q-主特征值,如果它有一个特征向量其分量的和不等于零.确定了所有恰有两个Q-主特征值的三圈图.     

双圈图的无符号拉普拉斯谱半径和谱展

The signless Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the smallest eigenvalue of its signless Laplacian matrix. In this paper, we determine the first to llth largest signless Laplacian spectral radii in the class of bicyclic graphs with n vertices. Moreover, the unique bicyclic graph with the largest or the second largest signless Laplacian spread among the class of connected bicyclic graphs of order n is determined, respectively.     

Signless Laplacian spectral radius and Hamiltonicity of graphs with large minimum degree

In this paper, we establish a tight sufficient condition for the Hamiltonicity of graphs with large minimum degree in terms of the signless Laplacian spectral radius and characterize all extremal graphs. Moreover, we prove a similar result for balanced bipartite graphs. Additionally, we construct infinitely many graphs to show that results proved in this paper give new strength for one to determine the Hamiltonicity of graphs.     

双圈图的Laplacian Spread(英文)

Laplacian spread的概念在刻画图的整体性质方面非常重要.近年来,Fan等分别刻画了树中具有极大和极小Laplacian spread的图.另外Bao等确定了在所有单圈图中具有极大Laplacian spread的图.边数减去顶点数目为1的连通图称为双圈图.令B_n是所有有n个顶点构成的双圈图集合.对n≥11,本文确定了B_n中所有具有极大Laplacian spread的那些图.     

Graphs with at most one signless Laplacian eigenvalue exceeding three

We determine all connected graphs with at most one signless Laplacian eigenvalue exceeding three.