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.     

The signless Laplacian spectral radius of tricyclic graphs and trees with k pendant vertices

In this paper, we consider the following problem: of all tricyclic graphs or trees of order n with k pendant vertices (n,k fixed), which achieves the maximal signless Laplacian spectral radius?We determine the graph with the largest signless Laplacian spectral radius among all tricyclic graphs with n vertices and k pendant vertices. Then we show that the maximal signless Laplacian spectral radius among all trees of order n with k pendant vertices is obtained uniquely at T_n,k, where T_n,k is a tree obtained from a star K_1,k and k paths of almost equal lengths by joining each pendant vertex to one end-vertex of one path. We also discuss the signless Laplacian spectral radius of T_n,k and give some results.     

A sharp upper bound for the spectral radius of a nonnegative matrix and applications

We obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of an undirected graph or a digraph. These results are new or generalize some known results.     

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.     

Distance signless Laplacian spectral radius and Hamiltonian properties of graphs

In this paper, we establish a sufficient condition on distance signless Laplacian spectral radius for a bipartite graph to be Hamiltonian. We also give two sufficient conditions on distance signless Laplacian spectral radius for a graph to be Hamilton-connected and traceable from every vertex, respectively. Furthermore, we obtain a sufficient condition for a graph to be Hamiltonian in terms of the distance signless Laplacian spectral radius of the complement of a graph G.     

Maximizing signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity

In this paper, we characterize the graphs with maximum signless Laplacian or adjacency spectral radius among all graphs with fixed order and given vertex or edge connectivity. We also discuss the minimum signless Laplacian or adjacency spectral radius of graphs subject to fixed connectivity. Consequently we give an upper bound of signless Laplacian or adjacency spectral radius of graphs in terms of connectivity. In addition we confirm a conjecture of Aouchiche and Hansen involving adjacency spectral radius and connectivity.     

哈密顿连通图和可迹图的新充分谱条件

令G是一个简单连通图,ρ(G)和q~D(G)分别为图G的邻接谱半径和距离无符号拉普拉斯谱半径.提供了图G是哈密顿连通的两个新的谱充分条件,这两个充分条件分别是以ρ(G)和q~D(G)表示的,其中G是G的补图.进一步地,还给出了以q~D(G)表示的图G是从任意一点出发都是可迹的新的谱充分条件,从而扩展和改进了文献中的结果.     

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).     

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.     

Some sufficient spectral conditions on Hamilton-connected and traceable graphs

In this paper, we establish some sufficient conditions for a graph to be Hamilton-connected in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph. Furthermore, we also give some sufficient conditions for a graph to be traceable from every vertex in terms of the edge number, the spectral radius and the signless Laplacian spectral radius.     

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.     

On the signless Laplacian index of unicyclic graphs with fixed diameter

In the paper, we identify graphs with the maximal signless Laplacian spectral radius among all the unicyclicgraphs with n vertices of diameter d.     

On the reduced signless Laplacian spectrum of a degree maximal graph

For a (simple) graph G, the signless Laplacian of G is the matrix A(G)+D(G), where A(G) is the adjacency matrix and D(G) is the diagonal matrix of vertex degrees of G; the reduced signless Laplacian of G is the matrix Δ(G)+B(G), where B(G) is the reduced adjacency matrix of G and Δ(G) is the diagonal matrix whose diagonal entries are the common degrees for vertices belonging to the same neighborhood equivalence class of G. A graph is said to be (degree) maximal if it is connected and its degree sequence is not majorized by the degree sequence of any other connected graph. For a maximal graph, we obtain a formula for the characteristic polynomial of its reduced signless Laplacian and use the formula to derive a localization result for its reduced signless Laplacian eigenvalues, and to compare the signless Laplacian spectral radii of two well-known maximal graphs. We also obtain a necessary condition for a maximal graph to have maximal signless Laplacian spectral radius among all connected graphs with given numbers of vertices and edges.     

强连通双圈有向图的无符号拉普拉斯谱刻画

设$\overrightarrow{G}$ 是一个强连通双圈有向图, $A(\overrightarrow{G})$是其邻接矩阵.设$D(\overrightarrow{G})$ 是$\overrightarrow{G}$的顶点出度的对角矩阵, $Q(\overrightarrow{G})=D(\overrightarrow{G})+A(\overrightarrow{G})$是$\overrightarrow{G}$ 的无符号拉普拉斯矩阵. $Q(\overrightarrow{G})$的谱半径称为$\overrightarrow{G}$的无符号拉普拉斯谱半径.在这篇文章中, 确定了在所有强连通双圈有向图中达到最大或最小无符号拉普拉斯谱半径的唯一有向图. 此外,还证明了任意一个强连通双圈有向图是由它的无符号拉普拉斯谱所确定的.     

The (signless) Laplacian spectral radius of unicyclic and bicyclic graphs with <Emphasis Type="Italic">n</Emphasis> vertices and <Emphasis Type="Italic">k</Emphasis> pendant vertices

In this paper, the effects on the signless Laplacian spectral radius of a graph are studied when some operations, such as edge moving, edge subdividing, are applied to the graph. Moreover, the largest signless Laplacian spectral radius among the all unicyclic graphs with n vertices and k pendant vertices is identified. Furthermore, we determine the graphs with the largest Laplacian spectral radii among the all unicyclic graphs and bicyclic graphs with n vertices and k pendant vertices, respectively.     

不含四圈的k圈图的无符号拉普拉斯谱半径（英文）

A k-cyclic graph is a connected graph of order n and size n + k-1. In this paper, we determine the maximal signless Laplacian spectral radius and the corresponding extremal graph among all C_4-free k-cyclic graphs of order n. Furthermore, we determine the first three unicycles and bicyclic, C_4-free graphs whose spectral radius of the signless Laplacian is maximal. Similar results are obtained for the(combinatorial) Laplacian     

Quotient of spectral radius, (signless) Laplacian spectral radius and clique number of graphs

In this paper, the upper and lower bounds for the quotient of spectral radius (Laplacian spectral radius, signless Laplacian spectral radius) and the clique number together with the corresponding extremal graphs in the class of connected graphs with n vertices and clique number ω(2 ≤ ω ≤ n) are determined. As a consequence of our results, two conjectures given in Aouchiche (2006) and Hansen (2010) are proved.     

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.     

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.     

无符号拉普拉斯谱半径的新上界

如果一个图存在定向满足其最大出度△~＋不超过最大度△的一半,则通过估计图的半边路径（semi-edge walk）的个数,得到了该图的无符号拉普拉斯谱半径的一个新上界.进而根据D.Goncalves对平面图边分解的结果,得到了平面图无符号拉普拉斯谱半径的一个新上界.