Some results on the ordering of the Laplacian spectral radii of unicyclic graphs

A unicyclic graph is a graph whose number of edges is equal to the number of vertices. Guo Shu-Guang [S.G. Guo, The largest Laplacian spectral radius of unicyclic graph, Appl. Math. J. Chinese Univ. Ser. A. 16 (2) (2001) 131–135] determined the first four largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices. In this paper, we extend this ordering by determining the fifth to the ninth largest Laplacian spectral radii together with the corresponding graphs among all unicyclic graphs on n vertices.     

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 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.     

The Laplacian spectral radius for unicyclic graphs with given independence number

In this paper, we give a complete characterization of the extremal graphs with maximal Laplacian spectral radius among all unicyclic graphs with given order and given number of pendent vertices. Then we study the Laplacian spectral radius of unicyclic graphs with given independence number and characterize the extremal graphs completely.     

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.     

单圈图的无符号狄利克雷谱半径

Let G be a simple connected graph with pendant vertex set ?V and nonpendant vertex set V_0. The signless Laplacian matrix of G is denoted by Q(G). The signless Dirichlet eigenvalue is a real number λ such that there exists a function f ≠ 0 on V(G) such that Q(G)f(u) = λf(u) for u ∈ V_0 and f(u) = 0 for u ∈ ?V. The signless Dirichlet spectral radiusλ(G) is the largest signless Dirichlet eigenvalue. In this paper, the unicyclic graphs with the largest signless Dirichlet spectral radius among all unicyclic graphs with a given degree sequence are characterized.     

On the Reverse Wiener Indices of Unicyclic Graphs

We determine the maximum values of the reverse Wiener indices of the unicyclic graphs with given cycle length, number of pendent vertices and maximum degree, respectively, and characterize the extremal graphs. We also determine the unicyclic graphs of given cycle length and diameter with minimum Wiener index.      

Remarks on the Wiener index of unicyclic graphs

Unicyclic graphs are connected graphs with the same number of vertices and edges. Tang and Deng (J. Math. Chem. 43:60–74, 2008) considered the problem of classification of all unicyclic graphs with the first three smallest and largest Wiener indices. In this paper, by construction of some classes of unicyclic graphs, the classification of unicyclic graphs with the first three maximum and minimum Wiener indices is completed.     

On the spectral radius of bicyclic graphs with n vertices and diameter d

Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. In this paper we determine the graph with the largest spectral radius among all bicyclic graphs with n vertices and diameter d. As an application, we give first three graphs among all bicyclic graphs on n vertices, ordered according to their spectral radii in decreasing order.     

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.     

给定阶与边独立数的树和单圈图的Laplacian矩阵的最大特征值

得到了给定顶点数和边独立数的树与单圈图的Laplacian矩阵的最大特征值的精确上界，并且给出了达到上界的所有极图．     

控制数固定的树的谱半径(英文)

阶数为n,控制数为γ的树的集合记为T_n,γ（其中n≥max{12,2γ+1},γ≥3）。本文给出了T_n,γ中前三大的邻接谱半径以及它们对应的图。     

The largest Wiener index of unicyclic graphs given girth or maximum degree

The Wiener index of a connected graph G is defined to be the sum of all distances of pairs of distinct vertices of G. Yu and Feng (Ars Comb 94:361–369, 2010) determined the unique graph having the largest Wiener index among all unicyclic graphs given girth. In the article we first give two new graphic transformations. Then with them we determine the graphs having the second largest Wiener index among all unicyclic graphs given girth and we also determine the graphs having the largest Wiener index among all unicyclic graphs given maximum degree.     

单圈共轭图的最小广义Randic指标R-1

图G的广义Randic指标定义为Rα=Rα（G）=∑uv∈E（G）（d（u）d（v））^α,其中d（u）是G的顶点u的度,α是任意实数．本文确定了单圈共轭图的广义Randic指标R-1的严格下界,并刻划了达到最小R-1的极图,这类极图还是化学图．     

单圈图的Laplace矩阵的最大特征值

利用阶数给出了单圈图的Laplace矩阵的最大特征值的第一,第二,第三,第四大值及最小值,并刻划达到上,下界的极图。     

单圈图的最大特征值序

主要讨论了单圈图按其最大特征值进行排序的问题，确定了该序的前六个图。     

图的无符号拉普拉斯和距离无符号拉普拉斯特征值

Let G be a simple graph. We first show that ■, where δiand di denote the i-th signless Laplacian eigenvalue and the i-th degree of vertex in G, respectively.Suppose G is a simple and connected graph, then some inequalities on the distance signless Laplacian eigenvalues are obtained by deleting some vertices and some edges from G. In addition, for the distance signless Laplacian spectral radius ρQ(G), we determine the extremal graphs with the minimum ρQ(G) among the trees with given diameter, the unicyclic and bicyclic graphs with given girth, respectively.     

Spectra of unicyclic graphs

Unicyclic graphs are discussed in the context of graph orderings related to eigenvalues. Several theorems involving lexicographical ordering by spectral moments as well as the ordering by the largest eigenvalue are proved. An appendix contains a table of the 89 unicyclic graphs on eight vertices together with their spectra, spectral moments and characteristic polynomials.Carnegie Research Fellow, University of Stirling, 1985–86.     

On the least eigenvalue of a unicyclic mixed graph

Using the result on Fiedler vectors of a simple graph, we obtain a property on the structure of the eigenvectors of a nonsingular unicyclic mixed graph corresponding to its least eigenvalue. With the property, we get some results on minimizing and maximizing the least eigenvalue over all nonsingular unicyclic mixed graphs on n vertices with fixed girth. In particular, the graphs which minimize and maximize, respectively, the least eigenvalue are given over all such graphs with girth 3.     

On the distance and distance signless Laplacian spectral radii of bicyclic graphs

The unique graphs with minimum and second-minimum distance (distance signless Laplacian, respectively) spectral radii are determined among bicyclic graphs with fixed number of vertices.