广义U—统计量的指数收敛速度

文「１」研究了一样本Ｕ－统计量的指数收敛速度,本文则研究了广义Ｕ－统计量的指数收敛速度。     

广义线性图嵌入分布的收敛速度

线性图序列的亏格分布已经被研究了30多年.此前,大部分文献关注于寻找图序列嵌入分布的具体表达式、递推关系式或者对数凹的证明.近期的研究表明:对于广义线性图序列■,在一定条件下,当n趋向于无穷时,■的嵌入分布会收敛于正态分布(参见[Adv.Appl.Math.,2021,127:102175].基于此工作,在类似的条件下,本文证明了其收敛速度的阶为■.同时,论证了该收敛速度估计的最优性.最后,给出了一些具体的例子.     

关于Γ-收敛的极限泛函表达式的注记

结合概周期函数的性质和Γ－收敛技巧，本文对很广泛的一类泛函的Γ－极限给出了统一的表达式，且此表达式比现在已有的结果更明确．     

关于λ-数乘收敛级数的不变性

本文证得（1）若λ具有弱滑脊性，那么λ-数乘收敛级数具有对偶不变性。（2）若λ包含C00，那么λ-数乘收敛级数具有全程不变性的充要条件为（λ，β（λ，λβ））是AK－空间。     

关于ρ-混合序列对数律的收敛速度

本文研究了ρ-混合序列对数律的收敛速度，在较弱的矩条件下得到了与独立同分布实随机变量类似的结果，并获得了ρ－混合序列满意对数律的一个充分性结果；讨论了ρ－混合序列重对数律的收敛速度的问题，得到了一个重对数律的充分性条件。     

关于收敛的P—级数和的近似值

当p>1时,p-级数sum from n≥1n 1/p是收敛的.若取p=2,是著名的Bernoulli级数.早在17世纪,由Euler用代数方程与三角函数方程进行类比的方法,给出了     

图的最大亏格与2-因子

图Ｇ的一个２因子Ｆ就是Ｇ的这样一个支撑子图，使其任何节点ｖ∈Ｖ的次ｄＦ（ｖ）＝２．易见，Ｇ的每个２因子均为无公共节点的圈之并．若Ｆ的每个圈的长均为３（或４），则称Ｇ含有一个三角形（或四边形）２因子．Ｍ．ｋ∨ｏｖｉｅｒａ［５］得到了含有三角形２因子的３－正则图的最大亏格．本文在３－正则图上，引进了扩张运算和讨论了与最大亏格和Ｂｅｔｉ亏数之间的关系．利用这些运算，得到了所有含四边形２因子的连通３－正则图是上可嵌入的，即γＭ（Ｇ）＝ｎ４（ｎ为Ｇ的节点数ｎ＝｜Ｖ（Ｇ）｜）．然后，基于此证明了含四边形２因子且所有节点ｖ∈Ｖ的次ｄＧ（ｖ）＝３（ｍｏｄ４）的图Ｇ均为上可嵌入的     

关于指数丢番图方程的Hugh Edgar问题

§1 Hall在组合数学的T型差集讨论中,提出了求解Diophantus方程p~m-q~n=2,m>1,n>1(p,g是素数) (1)的问题.后来,Hugh Edgar提出了更为一般的问题,即对给定的素数p,q和整数h,求方程     

一类密度估计的收敛速度

ＰｒａｋａｓａＲａｏ在文献［１］中提出一类密度估计ｆｎ（ｘ），我们得到当ｘ固定时ｆｎ（ｘ）－ｆ（ｘ）的ａ．ｓ．收敛速度及ｆｎ（ｘ）正态逼近的Ｂｅｒｒｙ－Ｅｓｓｅｅｎ界，同时，给出ｓｕｐｘ｜ｆｎ（ｘ）－ｆ（ｘ）｜的一致收敛速度     

Clique‐inverse graphs of K3‐free and K4‐free graphs

The clique graph K(G) of a given graph G is the intersection graph of the collection of maximal cliques of G. Given a family ℱ of graphs, the clique‐inverse graphs of ℱ are the graphs whose clique graphs belong to ℱ. In this work, we describe characterizations for clique‐inverse graphs of K₃‐free and K₄‐free graphs. The characterizations are formulated in terms of forbidden induced subgraphs. © 2000 John Wiley & Sons, Inc. J Graph Theory 35: 257–272, 2000     

Iterated clique graphs with increasing diameters

A simple argument by Hedman shows that the diameter of a clique graph G differs by at most one from that of K(G), its clique graph. Hedman described examples of a graph G such that diam(K(G)) = diam(G) + 1 and asked in general about the existence of graphs such that diam(Kⁱ(G)) = diam(G) + i. Examples satisfying this equality for i = 2 have been described by Peyrat, Rall, and Slater and independently by Balakrishnan and Paulraja. The authors of the former work also solved the case i = 3 and i = 4 and conjectured that such graphs exist for every positive integer i. The cases i ≥ 5 remained open. In the present article, we prove their conjecture. For each positive integer i, we describe a family of graphs G such that diam(Kⁱ(G)) = diam(G) + i. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 147–154, 1998     

On the Clique-Transversal Number in （Claw,K4）-Free 4-Regular Graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G.The clique-transversal number,denoted by τC(G),is the minimum cardinality of a clique-transversal set in G.In this paper,we first present a lower bound on τC(G) and characterize the extremal graphs achieving the lower bound for a connected(claw,K4)-free 4-regular graph G.Furthermore,we show that for any 2-connected(claw,K4)-free 4-regular graph G of order n,its clique-transversal number equals to [n/3].     

Centered graphs and the structure of ego networks

A way of comparing ego networks through examining patterns among their ties is introduced. It is derived from graph-theoretic ideas about centered graphs. An illustration using data from a computer conference is provided.     

Introducing subclasses of basic chordal graphs

Basic chordal graphs arose when comparing clique trees of chordal graphs and compatible trees of dually chordal graphs. They were defined as those chordal graphs whose clique trees are exactly the compatible trees of its clique graph.In this work, we consider some subclasses of basic chordal graphs, like hereditary basic chordal graphs, basic DV and basic RDV graphs, we characterize them and we find some other properties they have, mostly involving clique graphs.     

On graphs with the largest Laplacian index

Let G be a connected simple graph on n vertices. The Laplacian index of G, namely, the greatest Laplacian eigenvalue of G, is well known to be bounded above by n. In this paper, we give structural characterizations for graphs G with the largest Laplacian index n. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k-regular graph G of order n with the largest Laplacian index n. We prove that for a graph G of order n ⩾ 3 with the largest Laplacian index n, G is Hamiltonian if G is regular or its maximum vertex degree is Δ(G) = n/2. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results. The first author is supported by NNSF of China (No. 10771080) and SRFDP of China (No. 20070574006). The work was done when Z. Chen was on sabbatical in China.     

On a class of graphs without 3-stars

M. Numata described edge regular graphs without 3-stars. Allμ-subgraphs of these graphs are regular of the same valency. We prove that a connected graph without 3-stars all of whoseμ- subgraphs are regular of valencyα > 0 is either a triangular graph, or the Shläfli graph, or the icosahedron graph.     

Hamilton-connected indices of graphs

Let G be an undirected graph that is neither a path nor a cycle. Clark and Wormald [L.H. Clark, N.C. Wormald, Hamiltonian-like indices of graphs, ARS Combinatoria 15 (1983) 131-148] defined hc(G) to be the least integer m such that the iterated line graph L^m(G) is Hamilton-connected. Let be the diameter of G and k be the length of a longest path whose internal vertices, if any, have degree 2 in G. In this paper, we show that . We also show that κ³(G)≤hc(G)≤κ³(G)+2 where κ³(G) is the least integer m such that L^m(G) is 3-connected. Finally we prove that hc(G)≤|V(G)|−Δ(G)+1. These bounds are all sharp.     

Sphericity, cubicity, and edge clique covers of graphs

The sphericity sph(G) of a graph G is the minimum dimension d for which G is the intersection graph of a family of congruent spheres in R^d. The edge clique cover number θ(G) is the minimum cardinality of a set of cliques (complete subgraphs) that covers all edges of G. We prove that if G has at least one edge, then sph(G)?θ(G). Our upper bound remains valid for intersection graphs defined by balls in the L_p-norm for 1?p?∞.