Some Properties of Graphs of Diameters

The main result of this paper is as follows. Any two cycles of odd lengths of the graph of diameters G in three-dimensional Euclidean space have a common vertex. Some properties of graphs of diameters in two-dimensional Banach spaces with strictly convex metrics are also established. Applications are given. Received December 28, 1998, and in revised form September 10, 1999. Online publication May 15, 2000.     

关于高维单形的两个性质

应用度量几何理论与解析方法,研究了n维欧氏空间En中n维单形的性质,将三角形内心与中线两个性质推广到n维单形,获得n维单形内心与中位面的两个性质.     

关于单形的中面

在n维欧氏空间中,作为三角形的高维推广的单形的中面是最近才引入的一个重要的几何概念．该文利用Grassmann代数的方法获得了单形的中面面积的解析表达式,证明了单形的中面类似于三角形中线的性质,例如,对于一个给定的单形,存在另一个单形使得其边长分别等于给定单形的中面面积;一个单形的所有中面有且仅有一个公共点等．同时,利用中面面积的解析表达式证明了单形中面与单形的棱长、外接圆半径、中线长、角平分面等之间的一些优美性质,建立了一些新的重要的几何不等式．     

Diameters of octahedra

The Kolmogorov n-diameters of an m-dimensional octahedron O^m , i.e., of a convex hull of vectors {±₁e₁, ..., _me_m}, are calculated.Translated from Matematicheskie Zametki, Vol. 5, No. 4, pp. 429–436, April, 1969.In conclusion the author expresses his gratitude to V. M. Tikhomirov, who directed this work.     

Edgewise Subdivision of a Simplex

In this paper we introduce the abacus model of a simplex and use it to subdivide a d -simplex into k ^d d -simplices all of the same volume and shape characteristics. The construction is an extension of the subdivision method of Freudenthal [3] and has been used by Goodman and Peters [4] to design smooth manifolds. Received June 24, 1999, and in revised form January 13, 2000. Online publication August\/ 11, 2000.     

变更图的直径

P(t,n)和C(t,n)分别表示在阶为n的路和圈中添加t条边后得到的图的最小直径；f(t,k)表示从直径为k的图中删去t条边后得到的连通图的最大直径．这篇文章证明了t≥4且n≥5时,P(t,n)≤(n-8)/(t 1) 3；若t为奇数,则C(t,n)≤(n-8)/(t 1) 3；若t为偶数,则C(t,n)≤(n-7)/(t 2) 3．特别地,「(n-1)/5」≤P(4,n)≤「(n 3)/5」,「n/4」-1≤C(3,n)≤「n/4」．最后,证明了：若k≥3且为奇数,则f(t,k)≥(t 1)k-2t 4．这些改进了某些已知结果．     

变更图的直径

对于给定的正整数t和d(≥2),用F(t,d)和P(t,d)分别表示在所有直径为d的图和路中添加t条边后得到的图的最小直径,用f(t, d)表示从所有直径为d的图中删去t条边后得到的图的最大直径. 已经证明P(1, d)=(d)/(2), P(2,d)=(d 1)/(3)和P(3, d)=(d 2)/(4). 一般地,当t和d≥4时有(d 1)/(t 1)-1≤P(t, d)≤(d 1)/(t 1) 3. 在这篇文章中,我们得到F(t, f(t, d))≤d≤f(t, F(t, d))和(d)/(t 1)≤F(t, d)=P(t, d)≤(d-2)/(t 1) 3,而且当d充分大时,F(t, d)≤(d)/(t) 1. 特别地,对任意正整数k有P(t, (2k-1)(t 1) 1)=2k,当t=4或5,且d≥4时有(d)/(t 1)≤P(t, d)≤(d)/(t 1) 1.     

单形中的两个几何不等式(英文)

给出了n维欧氏空间中关于单形的n-1维体积与单形内部任意一点到所对界面的距离的两个不等式.     

Diameters of random circulant graphs

The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many small-world networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the worst connected networks are cycles whose diameters increase linearly in the number of nodes. In the present study we consider an intermediate class of examples: Cayley graphs of cyclic groups, also known as circulant graphs or multi-loop networks. We show that the diameter of a random circulant 2k-regular graph with n vertices scales as n ^1/k , and establish a limit theorem for the distribution of their diameters. We obtain analogous results for the distribution of the average distance and higher moments.     

Diameters of 3-sphere quotients

Let G⊂O(4) act isometrically on S³. In this article we calculate a lower bound for the diameter of the quotient spaces S³/G. We find it to be , which is exactly the value of the lower bound for diameters of the spherical space forms. In the process, we are also able to find a lower bound for diameters for the spherical Aleksandrov spaces, Sn/G, of cohomogeneities 1 and 2, as well as for cohomogeneity 3 (with some restrictions on the group type). This leads us to conjecture that the diameter of Sn/G is increasing as the cohomogeneity of the group G increases.     

Diameters of uniform subset graphs

Let r,s be positive integers with r>s, k a nonnegative integer, and n=2r−s+k. A uniform subset graph G(n,r,s) is a graph with vertex set [n]^r and where two r-subsets A,B∈[n]^r are adjacent if and only if |A∩B|=s. Let denote the diameter of a graph G.In this paper, we prove the following results: (1) If k>0, then if r≥2s+k+2, 2 if k≥s and 2s≤r≤s+k, or k<s and s+k≤r≤2s, and 3 otherwise; (2) If k=0, then . This generalizes a result in [M. Valencia-Pabon, J.-C. Vera, On the diameter of Kneser graphs, Discrete Math. 305 (2005) 383-385].     

The Genuine Bernstein-Durrmeyer Operator on a Simplex

In 1967 Durrmeyer introduced a modification of the Bernstein polynomials as a selfadjoint polynomial operator on L²[0,1] which proved to be an interesting and rich object of investigation. Incorporating Jacobi weights Berens and Xu obtained a more general class of operators, sharing all the advantages of Durrmeyer’s modification, and identified these operators as de la Vallée-Poussin means with respect to the associated Jacobi polynomial expansion. Nevertheless, all these modifications lack one important property of the Bernstein polynomials, namely the preservation of linear functions. To overcome this drawback a Bernstein-Durrmeyer operator with respect to a singular Jacobi weight will be introduced and investigated. For this purpose an orthogonal series expansion in terms generalized Jacobi polynomials and its de la Vallée-Poussin means will be considered. These Bernstein-Durrmeyer polynomials with respect to the singular weight combine all the nice properties of Bernstein-Durrmeyer polynomials with the preservation of linear functions, and are closely tied to classical Bernstein polynomials. Focusing not on the approximation behavior of the operators but on shape preserving properties, these operators we will prove them to converge monotonically decreasing, if and only if the underlying function is subharmonic with respect to the elliptic differential operator associated to the Bernstein as well as to these Bernstein-Durrmeyer polynomials. In addition to various generalizations of convexity, subharmonicity is one further shape property being preserved by these Bernstein-Durrmeyer polynomials. Finally, pointwise and global saturation results will be derived in a very elementary way.