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.     

关于双圈图的谱半径

如果G是连通的并且G的边数是n 1,那么n阶图G叫做双圈图,设B(n)是所有的阶为n的双圈图构成的集合,本文给出了B(n)(n(?)9)中前三大的邻接谱半径以及它们对应的图.     

双圈图按谱半径的排序

一个n阶简单连通图G被称为双圈图,如果它的边数是n+1.记B(n)是n阶双圈图的全体.本文确定了B(n)(n≥20)中谱半径的第六大至第十大值和对应的图.     

双圈图的距离无符号拉普拉斯谱半径

连通图$G$的距离无符号拉普拉斯矩阵定义为$\mathcal{Q}(G)=Tr(G)+D(G)$, 其中$Tr(G)$和$D(G)$分别为连通图$G$的点传输矩阵和距离矩阵. 图$G$的距离无符号拉普拉斯矩阵的最大特征值称为$G$的距离无符号拉普拉斯谱半径. 本文确定了给定点数的双圈图中具有最大的距离无符号拉普拉斯谱半径的图.     

图的Laplace谱半径的几类上界

本文得到图的Laplace谱半径的几类上界.通过选取适当的对角矩阵,我们得到了在一定程度上优于其他界的上界.     

连通图的最小Laplace谱半径的排序

首先找出了具有最小Laplace谱半径的第2个至第5个n阶单圈图和具有最小Laplace谱半径的n阶双圈图.然后结合有关n阶树的最小Laplace谱半径的排序,给出了所有n阶连通图中Laplace谱半径最小的14个图,当n为偶数时,它们达到了所有佗阶连通图中Laplace谱半径最小的9个值(其中有并列的),而当n为奇数时,它们则达到了Laplace谱半径最小的8个值(其中有并列的).     

有向图的Laplace谱半径

Laplace矩阵的谱半径一直是近年来谱图理论的研究热点.本文主要讨论有向图Laplace矩阵的谱半径,用顶点的出度和公共邻域数给出了谱半径上界,用图的最大出度给出了一些特殊图类谱半径的下界.     

单圈图与双圈图补图的Aα-谱半径

设A(G)和D(G)分别表示n阶图G的邻接矩阵和度对角矩阵,对于任意实数α∈[0,1],图G的Aα-矩阵被定义为Aα(G)=αD(G)+(1?α)A(G),它是图的邻接矩阵和无符号拉普拉斯矩阵的共同推广,其最大特征根称为图G的Aα-谱半径.单圈图与双圈图补图的Aα-谱半径的上界被分别确定,相应的极图被完全刻画.     

图的点覆盖数与其Laplace谱半径

本文讨论图的点覆盖数与图的 Laplace谱半径的关系 ,利用特征向量的技巧得到由图的 L aplace谱半径所确定的关于图的点覆盖数的紧的界     

恰有k个悬挂点的n阶双圈和三圈图的拟拉普拉斯谱半径

设k,n为两个确定的正整数.本文得到了当1≤k≤n-7时恰有k个悬挂点的n阶连通三圈图的最大拟拉普拉斯谱半径的唯一极图,也得到了当1≤k≤n-5时恰有k个悬挂点的n阶连通双圈图的最大拟拉普拉斯谱半径的唯一极图.     

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).     

不含三圈的k圈图的拟拉普拉斯和拉普拉斯谱半径

k圈图是边数等于顶点数加k-1的简单连通图.文中确定了不含三圈的k圈图的拟拉普拉斯谱半径的上界,并刻画了达到该上界的极图.此外,文中确定了拟拉普拉斯谱半径排在前五位的不含三圈的单圈图,排在前八位的不含三圈的双圈图.最后说明文中所得结论对不含三圈的k圈图的拉普拉斯谱半径也成立.     

On the Laplacian spectral radii of bicyclic graphs

A graph G of order n is called a bicyclic graph if G is connected and the number of edges of G is n+1. Let B(n) be the set of all bicyclic graphs on n vertices. In this paper, we obtain the first four largest Laplacian spectral radii among all the graphs in the class B(n) (n≥7) together with the corresponding graphs.     

三圈图的无符号拉普拉斯谱半径

假设图G的点集是V(G)={v_1,v_2,…,v_n},用d_(v_i)(G)表示图G中点v_i的度,令A(G)表示G的邻接矩阵,D(G)是对角线上元素等于d_(v_i)(G)的n×n对角矩阵,Q(G)=D(G)+A(G)是G的无符号拉普拉斯矩阵,Q(G)的最大特征值是G的无符号拉普拉斯谱半径.现确定了所有点数为n的三圈图中无符号拉普拉斯谱半径最大的图的结构.     

The Laplacian spectral radius of graphs on surfaces

Let G be an n-vertex (n?3) simple graph embeddable on a surface of Euler genus γ (the number of crosscaps plus twice the number of handles). Denote by Δ the maximum degree of G. In this paper, we first present two upper bounds on the Laplacian spectral radius of G as follows:

(i)

     

Some results on Laplacian spectral radius of graphs with cut vertices

In this paper, we give some results on Laplacian spectral radius of graphs with cut vertices, and as their applications, we also determine the unique graph with the largest Laplacian spectral radius among all unicyclic graphs with n vertices and diameter d, 3?d?n−3.     

The signless Laplacian spectral radius of graphs with given degree sequences

In this paper, we investigate the properties of the largest signless Laplacian spectral radius in the set of all simple connected graphs with a given degree sequence. These results are used to characterize the unicyclic graphs that have the largest signless Laplacian spectral radius for a given unicyclic graphic degree sequence. Moreover, all extremal unicyclic graphs having the largest signless Laplacian spectral radius are obtained in the sets of all unicyclic graphs of order n with a specified number of leaves or maximum degree or independence number or matching number.     

单圈图和双圈图的最大无符号拉普拉斯分离度

设G是一个n阶简单图,q_{1}(G)\geq q_{2}(G)\geq \cdots \geq q_{n}(G)是其无符号拉普拉斯特征值. 图G的无符号拉普拉斯分离度定义为S_{Q}(G)=q_{1}(G)-q_{2}(G). 确定了n阶单圈图和双圈图的最大的无符号拉普拉斯分离度,并分别刻画了相应的极图.