均匀递归树的去点问题(英文)

本文研究了均匀递归树的去点过程.对于任意给定的点i,利用矩母函数,给出了对应子树顶点数的准确分布,推广了A.Panholzer所做的工作.     

一类最小覆盖问题

对于任意给定的平行四边形,确定了与其相似的直径为1的三角形集族的最小平行四边形覆盖.     

给定最大匹配数的树的零阶广义Randid(c)指标

图G的零阶广义Randi指标定义为0Rα（G）=v∈V（G）d（v）α,其中d（v）为图G的顶点v的度,α为任意实数.研究了树的零阶广义Rα指标的极值问题,利用分析和图的理论,确定了任意给定最大匹配数的树的最大和最小Rα的值,并刻画了达到该极值的树.     

瞬态接触动力学的三维问题

发展了三维瞬态接触动力学问题的分析解法,并且得到了广义Galin()定理.对于给定的相当普遍的接触应力,可以确定相应的接触位移,并且得到解析的形式;反过来,对于给定的相当普遍的接触位移,可以解析地确定其相应的接触应力.上述所指的普遍的给定物理量可以是X与y的任意阶多项式和时间t的任意瞬态函数.还针对不同接触区的几何特征和最典型的最常使用的7种固体介质进行了实例计算,所有计算结果均表明上述理论结果在数学上和物理上的精确性.     

曲面上图的拉普拉斯谱半径

本文研究了图嵌入到给定紧致曲面上的拉普拉斯谱半径,确定了将顶点数为n、最大度为△的图分别嵌入到亏格为g的定向曲面和亏格为h的不可定向曲面上的新上界.     

禁用C4的IC-可平面图中Ck的最大个数

给定图G和H,如果图G不包含图H作为子图,则称图G是禁用H的.用ex_ICP(n,F,H)表示在所有禁用H且顶点数为n的IC-可平面图中,含与F同构的子图的最大个数.本文证明了对任意的k≥5,当n充分大时,有ex_ICP(n,C_k,C₄)=Θ(■).     

图的平面Turán数和平面anti-Ramsey数（英文）

在所有顶点数为n且不包含图G作为子图的平面图中,具有最多边数的图的边数称为图G的平面Turán数,记为ex_P(n,G)。给定正整数n以及平面图H,用T_n(H)来表示所有顶点数为n且不包含H作为子图的平面三角剖分图所组成的图集合。设图集合T_n (H)中的任意平面三角剖分图的任意k边染色都不包含彩虹子图H,则称满足上述条件的k的最大值为图H的平面anti-Ramsey数,记作ar_P(n,H)。两类问题的研究均始于2015年左右,至今已经引起了广泛关注。全面地综述两类问题的主要研究成果,以及一些公开问题。     

具有相同边界的单圈图的Faber-Krahn型不等式}

通过研究具有相同``体积'的图类中具有最小的第一Dirichlet特征值的图的性质, 刻画了具有给定边界点数的单圈图中具有Faber-Krahn性质的图.     

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.     

λ_5-geometry中的Steiner树问题(Ⅰ)

本文首先提出了 λ5-geometry中的 Steiner最小树问题 .讨论了 λ5-ge-ometry中的 Steiner最小树的若干性质 ,并给出了给定点数为 3或 4时 Steiner最小树的基本结构 .     

阶、时滞及系数都未知的系统的辨识和适应控制

假设对一个客观的物理系统我们一无所知,唯一可以获得的信息是给了系统的输入后可以量测到系统的输出、对这种“一无所知”的系统的辨识和适应控制并不是容易的事情,下面我们假定客观的系统是由随机差分方程描述的,即输入、输出和噪声之间的关系如下:     

The Estrada index of trees

The Estrada index of a graph G is defined as , where λ₁,λ₂,…,λ_n are the eigenvalues of its adjacency matrix. We determine the unique tree with maximum Estrada index among the set of trees with given number of pendant vertices. As applications, we determine trees with maximum Estrada index among the set of trees with given matching number, independence number, and domination number, respectively. Finally, we give a proof of a conjecture in [J. Li, X. Li, L. Wang, The minimal Estrada index of trees with two maximum degree vertices, MATCH Commun. Math. Comput. Chem. 64 (2010) 799-810] on trees with minimum Estrada index among the set of trees with two adjacent vertices of maximum degree.     

k-分支星状树的第二几何算术指标

数学化学中,第二几何算术指标是新近提出的一个图的拓扑指标,它与Szeged指标和点PI指标具有紧密关系.如果树的一个顶点υ的度大于等于3,则称顶点υ是其一个分支点.通过树的第二几何算术指标的一个增加或减少的变换,刻画了k-分支星状树的第二几何算术指标的最值,同时确定了相应的极图.     

Elementary Properties of Cycle-free Partial Orders and their Automorphism Groups

A classification was given by Creed, Truss, and Warren of all the countable k-CS-transitive cycle-free partial orders for k3. Here the elementary theories of these structures and their automorphism groups are examined, and it is shown that in many cases we can distinguish the structures or their groups by means of their first- or second-order properties. The small index property is established for weakly 2-transitive trees, and for several classes of cycle-free partial orders.     

Corrections of proofs for Hansen and Mélot's two theorems

The Randi? index R(G) of a (chemical) graph G is also called connectivity index. Hansen and Mélot [Variable neighborhood search for extremal graphs 6: analyzing bounds for the connectivity index, J. Chem. Inf. Comput. Sci. 43 (2003) 1-14] in their paper, characterized the chemical trees of given order and number of pendent vertices which have the minimum and maximum Randi? index, respectively. In this note, we point out the mistakes in the proofs of their results Theorems 8 and 10, while we still believe that the two theorems are true, and then we give their corrected proofs.     

The Number of Subtrees of Trees with Given Degree Sequence

This article investigates some properties of the number of subtrees of a tree with given degree sequence. These results are used to characterize trees with the given degree sequence that have the largest number of subtrees, which generalize the recent results of Kirk and Wang (SIAM J Discrete Math 22 (2008), 985–995). These trees coincide with those which were proven by Wang and independently Zhang et al. (2008) to minimize the Wiener index. We also provide a partial ordering of the extremal trees with different degree sequences, some extremal results follow as corollaries.     

On the binary projective codes with dimension 6

All binary projective codes of dimension up to 6 are classified. Information about the number of the codes with different minimum distances and automorphism group orders is given.     

通过增强型萨格勒布指数对图排序

Recently, Furtula et al. proposed a valuable predictive index in the study of the heat of formation in octanes and heptanes, the augmented Zagreb index(AZI index) of a graph G, which is defined as AZI(G) =∑uv∈E(G)( d_u d_v/d_u + d_v-2)~3,where E(G) is the edge set of G, d u and d v are the degrees of the terminal vertices u and v of edge uv, respectively. In this paper, we obtain the first five largest(resp., the first two smallest) AZI indices of connected graphs with n vertices. Moreover, we determine the trees of order n with the first three smallest AZI indices, the unicyclic graphs of order n with the minimum, the second minimum AZI indices, and the bicyclic graphs of order n with the minimum AZI index, respectively.     

Totally interpolating biorthogonal multiwavelet systems with approximation order

In this article, compactly supported totally interpolating biorthogonal multiwavelet systems are studied. Necessary and sufficient conditions for such systems to have given approximation orders are stated in simple equations. It is shown that the shorter nontrivial filter component that has the minimum possible length for a given approximation order is uniquely determined up to a discrete parameter. Among systems with such property, we provide all totally interpolating biorthogonal stable multiwavelet systems of approximation orders 2 and 3 with minimal total length whose scaling vectors have minimal lengths as well.