消去图、覆盖图和均匀图的若干结果

设 G是一个图 ,g,f是定义在图 G的顶点集上的两个整数值函数 ,且g≤f.图 G的一个 ( g,f) -因子是 G的一个支撑子图 F,使对任意的 x∈V( F)有g( x)≤ d F( x)≤ f ( x) .文中推广了 ( g,f) -消去图、( g,f ) -覆盖图和 ( g,f) -均匀图的概念 ,给出了在 g相似文献   

书本图Bm的协调和相继标号

本文给出了书本图Ｂ２ｍ的ｍ种不同的相继标号和Ｂ４ｍ＋１的ｍ种不同的相继标号。因而，书本图Ｂｍ是相继图的充分条件为：ｍ＞１且ｍ≠（３ｍｏｄ４）。这一条件也是书本图Ｂｍ是协调图的充要条件。     

（mg＋m—1,mf—m＋1）—图的（g,f）—因子

本文证明了（ｍｇ＋ｍ－１,ｍｆ－ｍ＋１）－图具有一些特殊的（ｇ,ｆ）－因子,从而推广到了关于（ｇ,ｆ）－覆盖图和（ｇ,ｆ）－消去图的有关结果,有助于进一步研究（ｍｇ＋ｍ－１,ｍｆ－ｍ＋１）－图的正交因子分解问题。     

循环图Cn〈j1,j2,…,jr,n／2〉的连通性

本文得到了奇数度循环图C_n是连通的充要条件及C_n不连通的情形.证明了三度连通循环图C_n同构于C_n<1,n/2>或C_n<2,n/2>.这一结果颇有意义.     

Hamiltonian二部竞赛图

设 T(n,n)表示 n×n 二部竞赛图。本文证明了:如果 uv 是 T(n,n)的一条弧,蕴含d~-(u) d~ (v)≥n-2≥4,则 T(n,n)是 Hamilton 图,除非 T(n,n)属于两类已被刻划的特殊图类。     

(g,f)-消去图的若干充分条件

设 G是一个图 ,用 V(G)和 E(G)表示它的顶点集和边集 ,并设 g和 f是定义在 V(G)上的两个整数值函数且 g 相似文献   

Hamilton图的特定生成了图问题的反例

［１］定理３断言：一个Ｈａｍｉｌｔｏｎ图Ｇ必存在仅有ｐ条桥的相间偶圈，如果相间偶圈的边中有边在Ｇ的ｐ个不连通初等子圈上（ｐ≥２）。本的反例表明上述结论是错的，从而［１］中关于Ｐｅｔｅｒｓｏｎ图不是Ｈａｍｉｌｔｏｎ图的证明也不成立。     

（a,b,k）—临界图

设Ｇ是一个图且设ａ，ｂ是非负整数，ａ〈ｂ。如果消去Ｇ的任意ｋ个顶点剩下的图有〔ａ，ｂ〕因子，则称图Ｇ是（ａ，ｂ，ｋ）－临界图。本文给出了一个图是（ａ，ｂ，ｋ）－临界图的一个充分必要条件。讨论了该条件的一些应用，研究了（ａ，ｂ，ｋ）－临界图的性质。     

两类满足A（H）=3的图

本文利用文[3]的方法给出了两类满足A(H)=3的图,从而肯定了满足A(H)=3的图是不唯一的。本文还给出了满足A(H)=3的最小图。     

On the second largest Laplacian eigenvalues of graphs

The second largest Laplacian eigenvalue of a graph is the second largest eigenvalue of the associated Laplacian matrix. In this paper, we study extremal graphs for the extremal values of the second largest Laplacian eigenvalue and the Laplacian separator of a connected graph, respectively. All simple connected graphs with second largest Laplacian eigenvalue at most 3 are characterized. It is also shown that graphs with second largest Laplacian eigenvalue at most 3 are determined by their Laplacian spectrum. Moreover, the graphs with maximum and the second maximum Laplacian separators among all connected graphs are determined.     

The smallest eigenvalue of the signless Laplacian

Recently the signless Laplacian matrix of graphs has been intensively investigated. While there are many results about the largest eigenvalue of the signless Laplacian, the properties of its smallest eigenvalue are less well studied. The present paper surveys the known results and presents some new ones about the smallest eigenvalue of the signless Laplacian.     

On the Laplacian Eigenvalues of Signed Graphs

A signed graph is a graph with a sign attached to each edge. This article extends some fundamental concepts of the Laplacian matrices from graphs to signed graphs. In particular, the largest Laplacian eigenvalue of a signed graph is investigated, which generalizes the corresponding results on the largest Laplacian eigenvalue of a graph.     

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.     

On the extremal values of the second largest Q-eigenvalue

We study extremal graphs for the extremal values of the second largest Q-eigenvalue of a connected graph. We first characterize all simple connected graphs with second largest signless Laplacian eigenvalue at most 3. The second part of the present paper is devoted to the study of the graphs that maximize the second largest Q-eigenvalue. We construct families of such graphs and prove that some of theses families are minimal for the fact that they maximize the second largest signless Laplacian eigenvalue.     

Some results on the Laplacian eigenvalues of unicyclic graphs

In this paper, we provide the smallest value of the second largest Laplacian eigenvalue for any unicyclic graph, and find the unicyclic graphs attaining that value. And also give an “asymptotically good” upper bounds for the second largest Laplacian eigenvalues of unicyclic graphs. Using this results, we can determine unicyclic graphs with maximum Laplacian separator. And unicyclic graphs with maximum Laplacian spread will also be determined.     

The minimum Laplacian spread of unicyclic graphs

The Laplacian spread of a graph [1] is defined as the difference between the largest eigenvalue and the second-smallest eigenvalue of the associated Laplacian matrix. In this paper, the minimum Laplacian spread of unicyclic graphs with given order is determined.     

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.     

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 smallest signless Laplacian spectral radius of graphs with a given clique number

The signless Laplacian spectral radius of a graph G is the largest eigenvalue of its signless Laplacian matrix. In this paper, the first four smallest values of the signless Laplacian spectral radius among all connected graphs with maximum clique of size greater than or equal to 2 are obtained.