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.     

图的各类矩阵表示的谱半径的界（英文）

本文首先借助正尺度向量得到了非负不可约矩阵谱半径的界,结果改进和推广了部分已知结果.更进一步地,应用所得结果到图G的各类矩阵表示,得到了图G的带有参数α的各类谱半径的新上下界,这些结果理论上改进和推广了一些已有结果.     

非负矩阵与有向图的谱半径

本文给出非负矩阵的谱半径的上界、下界,由此给出有向图的谱半径的界.     

一致超图谱半径界的改进结果

利用张量理论研究一致超图的谱半径.首先,利用对角相似张量与原张量同谱的性质,结合张量特征值的圆盘定理,给出谱半径的上界,这一上界严格小于最大度;其次,通过超图的度向量给出谱半径的下界.改进了超图谱半径上下界的原有结果.     

不含三角形图的$A_\alpha$-谱半径的界

设$G$是简单无向图. 对于实数$\alpha \in [0,1]$, Nikiforov于2017年定义图的$A_\alpha$-矩阵为$A_\alpha(G)=\alpha D(G)+(1-\alpha)A(G)$, 其中$A(G)$和$D(G)$分别为图$G$的邻接矩阵和度对角矩阵. 图的$A_\alpha$-矩阵可以看着是图的邻接矩阵和无符号拉普拉斯矩阵的共同推广, 其最大特征值称为图的$A_\alpha$- 谱半径. 对于$\alpha\in[0,1)$, 本文确定了不含三角形图的$A_\alpha$-谱半径的一个下界;对于$\alpha \in[1/2, 1)$, 本文确定了不含三角形$k$圈图的$A_\alpha$-谱半径的一个上界.     

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).     

围长为g且有k个悬挂点的单圈图的谱半径和Laplacian谱半径

图的谱半径和Laplacian谱半径分别是图的邻接矩阵和Laplacian矩阵的最大特征值.本文中,我们分别刻画了围长为g且有k个悬挂点的单圈图的谱半径和Laplacian谱半径达到最大时的极图.     

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.     

张量A和A~k的Z-特征值的关系及其应用

给出了张量A和A的k次幂A~k的Z-特征值的关系,作为应用,给出了弱对称非负张量Z-谱半径的新下界估计式,改进了某些已有结果.     

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.     

The new upper bounds on the spectral radius of weighted graphs

Let us consider weighted graphs, where the weights of the edges are positive definite matrices. The eigenvalues of a weighted graph are the eigenvalues of its adjacency matrix and the spectral radius of a weighted graph is also the spectral radius of its adjacency matrix. In this paper, we obtain two upper bounds for the spectral radius of weighted graphs and compare with a known upper bound. We also characterize graphs for which the upper bounds are attained.     

Lower Bounds on the (Laplacian) Spectral Radius of Weighted Graphs

The weighted graphs, where the edge weights are positive numbers, are considered. The authors obtain some lower bounds on the spectral radius and the Laplacian spectral radius of weighted graphs, and characterize the graphs for which the bounds are attained. Moreover, some known lower bounds on the spectral radius and the Laplacian spectral radius of unweighted graphs can be deduced from the bounds.     

Extremal graph characterization from the bounds of the spectral radius of weighted graphs

We consider weighted graphs, where the edge weights are positive definite matrices. The eigenvalues of a graph are the eigenvalues of its adjacency matrix. We obtain a lower bound and an upper bound on the spectral radius of the adjacency matrix of weighted graphs and characterize graphs for which the bounds are attained.     

Sharp upper and lower bounds for the Laplacian spectral radius and the spectral radius of graphs

In this paper, sharp upper bounds for the Laplacian spectral radius and the spectral radius of graphs are given, respectively. We show that some known bounds can be obtained from our bounds. For a bipartite graph G, we also present sharp lower bounds for the Laplacian spectral radius and the spectral radius, respectively.     

非正则图的最大特征值

通过对图的最大特征分量与顶点度之间的关系的刻画,得到了图的谱半径与参数最大度和次大度之间的不等关系,进而获得了简单连通非正则图的谱半径的若干上界.     

Sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix and its applications

In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known results about various spectral radii, including the adjacency spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph.     

The Laplacian spectral radius of graphs

The Laplacian spectral radius of a graph is the largest eigenvalue of the associated Laplacian matrix. In this paper, we improve Shi’s upper bound for the Laplacian spectral radius of irregular graphs and present some new bounds for the Laplacian spectral radius of some classes of graphs.     

The spectral radii of a graph and its line graph

The spectral radius of a (directed) graph is the largest eigenvalue of adjacency matrix of the (directed) graph. We give the relation on the characteristic polynomials of a directed graph and its line graph, and obtain sharp bounds on the spectral radius of directed graphs. We also give the relation on the spectral radii of a graph and its line graph. As a consequence, the spectral radius of a connected graph does not exceed that of its line graph except that the graph is a path.     

Sharp bounds on distance spectral radius of graphs

Let D(G) denote the distance matrix of a connected graph G. The largest eigenvalue of D(G) is called the distance spectral radius of a graph G, denoted by ?(G). In this article, we give sharp upper and lower bounds for the distance spectral radius and characterize those graphs for which these bounds are best possible.     

Bounds on the eigenvalues of graphs with cut vertices or edges

In this paper, we characterize the extremal graph having the maximal Laplacian spectral radius among the connected bipartite graphs with n vertices and k cut vertices, and describe the extremal graph having the minimal least eigenvalue of the adjacency matrices of all the connected graphs with n vertices and k cut edges. We also present lower bounds on the least eigenvalue in terms of the number of cut vertices or cut edges and upper bounds on the Laplacian spectral radius in terms of the number of cut vertices.