树,单圈图,双圈图依谱矩的排序

设G=(V(G),E(G))是一个n阶简单图,V(G),E(G)分别为图G的顶点集和边集.G的k阶谱矩sk(G)为G的所有特征值λ1,λ2,···,λn的k次幂之和,即sk(G)=n i=1λi k.该文首先列出图的五种变换,然后得到了其对任意图的零到四阶谱矩的变化规律,最后依次给出了树和单圈图依谱矩序列S4的字典序分别排在前4-6位和后4-6的图及其特征以及双圈图依谱矩序列S4的字典序排在前6位和后6位的图及其特征.     

树和单圈图的斜谱矩

给定图$G$,对图$G$的每条边确定一个方向,称为$G$的定向图$G^\sigma$, $G$称为$G^\sigma$的基础图. $G^\sigma$的斜邻接矩阵$S(G^\sigma)$是反对称矩阵,其特征值是0或纯虚数. $S(G^\sigma)$所有特征值的$k$次幂之和称为$G^\sigma$的$k$阶斜谱矩,其中$k$是非负整数.斜谱矩序列可用于对图进行排序.本文主要研究定向树和定向单圈图的斜谱矩,并对这两类图的斜谱矩序列依照字典序进行排序.首先确定了直径为$d$的树作为基础图的所有定向树中,斜谱矩序最大的$2\lfloor\frac{d}{4}\rfloor$个图; 然后确定以围长为$g$的单圈图作为基础图的所有定向单圈图中, 斜谱矩序最大的$2\lfloor\frac{g}{4}\rfloor+1$个图.     

一个矩的不等式及其应用

本文证明了一类随机变量数学期望关于参数的单调性，并指出它在序贯分析中的应用。     

小波变换与矩心

利用多分辨分析的思想,研究了信号经波波变换后矩心的计算问题。     

树的Nordhaus-Gaddum类型谱半径的排序

给出了n阶树的Nordhaus-Gaddum类型谱半径即图及其补图的谱半径之和的可达上界:ρ(T) ρ(Tc)≤■ n-2,等号成立当且仅当T K1,n-1,其中Tc为T的补图,K1,n-1为n阶星图.同时证明了对于n阶双星图S(a,b)的Nordhaus-Gaddum类型谱半径随a的值单调上升,其中[n-1/2]≤a≤n-3.     

对角双线性时间序列的近似三阶矩结构和双谱密度

三阶矩结构和双谱是分析和研究非线性时间序列的重要方法。但对于双线性时间序列来说，计算两者的精确值是困难的。因此，寻找比较好的近似值就非常有必要。本文针对一般的对角双线性时间序列，给出了计算近似三阶矩和双谱密度的公式。仿真计算表明，这些公式是有效的。     

随机图的谱矩

简单图G的k阶谱矩定义为G的特征值的k阶幂之和,记为Mk(G).应用概率和代数的方法,对于几乎所有的图G,本文给出Mk(G)的一个精确估计.此外,对于几乎所有的多部图G,本文给出了Mk(G)的上界和下界.     

关于总体分布的矩确定

本文引进了局部分布、局部矩、局部矩法估计、样本局部矩和局部分布的矩确定等概念。从而为著名的经典矩量问题提供了新的研究途径与方法并获得了若干新结果。     

Ordering graphs with cut edges by their spectral radii

Let Gnk denote a set of graphs with n vertices and k cut edges. In this paper, we obtain an order of the first four graphs in Gnk in terms of their spectral radii for 6 ≤ k ≤ (n-2)/3.     

Ordering trees by their spectral radii

Let T be a tree with n vertices and let A（T） be the adjacency matrix of T. Spectral radius of T is the largest eigenvalue of A（T）. Wu et al. [Wu, B.F., Yuan, X.Y, and Xiao, E.L. On the spectral radii of trees, Journal of East China Normal University （Natural Science）, 3：22-28 （2004）] determined the first seven trees of order n with the smallest spectral radius. In this paper, we extend this ordering by determining the trees with the eighth to the tenth smallest spectral radius among all trees with n vertices.     

Sharp bounds for Laplacian spectral moments of digraphs with a fixed dichromatic number

The k-th Laplacian spectral moment of a digraph G is defined as ${\sum }_{i=1}^n{\lambda }_i^k$, where ${\lambda }_i$ are the eigenvalues of the Laplacian matrix of G and k is a nonnegative integer. For $k=2$, this invariant is better known as the Laplacian energy of G. We extend recently published results by characterizing the digraphs which attain the minimal and maximal Laplacian energy within classes of digraphs with a fixed dichromatic number. We also determine sharp bounds for the third Laplacian spectral moment within the special subclass which we define as join digraphs. We leave the full characterization of the extremal digraphs for $k\ge 3$ as an open problem.     

Signless Laplacian spectral characterization of line graphs of T-shape trees

A T-shape tree is a tree with exactly one vertex of maximum degree 3. The line graphs of the T-shape trees are triangles with a hanging path at each vertex. Let C_a,b,c be such a graph, where a, b and c are the lengths of the paths. In this paper, we show that line graphs of T-shape trees, with the sole exception of C_a,a,2a+1, are determined by the spectra of their signless Laplacian matrices. For the graph C_a,a,2a+1 we identify the unique non-isomorphic graph sharing the same signless Laplacian characteristic polynomial.     

Connected domination critical graphs with respect to relative complements

A dominating set in a graph G is a connected dominating set of G if it induces a connected subgraph of G. The minimum number of vertices in a connected dominating set of G is called the connected domination number of G, and is denoted by γ _c (G). Let G be a spanning subgraph of K _s,s and let H be the complement of G relative to K _s,s ; that is, K _s,s = G ⊕ H is a factorization of K _s,s . The graph G is k-γ _c -critical relative to K _s,s if γ _c (G) = k and γ _c (G + e) < k for each edge e ∈ E(H). First, we discuss some classes of graphs whether they are γ _c -critical relative to K _s,s . Then we study k-γ _c -critical graphs relative to K _s,s for small values of k. In particular, we characterize the 3-γ _c -critical and 4-γ _c -critical graphs.