广义de Bruijn和Kautz有向图的距离控制数

对于任意的正整数(?),强连通图G的顶点子集D被称为距离(?)-控制集,是指对于任意顶点v(?)D,D中至少含有一个顶点u,使得距离dG(u,v)≤(?)．图G距离(?)- 控制数γe(G)是指G中所有距离(?)-控制集的基数的最小者．本文给出了广义de Bruijn 和广义Kautz有向图的距离(?)-控制数的上界和下界,并且给出当它们的距离2-控制数达到下界时的一个充分条件．从而得到对于de Bruijn有向图B(d,k)的距离2-控制数γ2(B(d,k))= ．在该文结尾,我们猜想Kautz有向图K(d,k)的距离2-控制数γ2(K(d,k))= .     

关于超立方体网络的(d,k)独立数

(d,k)独立数是分析互连网络性能的一个重要参数.对于任意给定的图G和正整数d和k,确定G的(d,k)独立数问题是一个NPC问题.因此,确定一些特殊图的(d,k)独立数显得很重要.本文确定了k维超立方体网络的(d,k)独立数等于2,如果d=k≥4或者d=k-1≥6 以及αd,k-t(Qk)=αd,k(Qk),其中0≤t≤k-2,1≤d≤k-t-1.     

赋权图中的路和圈

本文研究了赋权图中的最长路和最长圈,将关于非赋权图中最长路和最长圈的一些结果推广到赋权图上.     

网络环境下数学课堂教学的设计与思考——“魔灯”网络教学

2006年,“魔灯”被引入学校,在上海师范大学教育技术系教授和研究生的指导下,我们开始探讨“魔灯”在教学过程中的应用,成立了“魔灯”研究课题组．笔者以“函数y=Asin（ωx＋ψ）的图像与性质应用”为例探讨网络环境下的魔灯课堂教学设计．     

环的左(N,U)-凝聚维数和Excellent扩张

Let S be an excellent extension of a ring R and U a flat right S-module. We show in this paper that the left (N,U)-coherent dimension of S is equal to that of R.     

Kautz网络中的最小反馈点集

对简单有向图D=(V，E)，顶点子集F∈V，如果由V＼F导出的子图不含有向圈，则称F是D的反馈点集。点数最小的子集F称为最小反馈点集。最小的点数称为反馈数。本利用Kautz最小轨道的方法确定出了Kautz有向图K(d，k)反馈数的一个下界和上界。并且具体给出了当k≤3时的反馈数。     

马尔可夫骨架PERT网络的最长路径

本文利用马尔可夫骨架过程理论研究PERT网络模型，其中网络各弧线的长度是相互独立的随机变量。文中构造了一个马尔可夫骨架过程，利用其向后方程求解随机网络最长路径长度的分布函数。     

无向Kautz图的超级限制边连通性

限制边连通度作为边连通度的推广,是计算机互连网络可靠性的一个重要度量.Superλ-′是比限制边连通度更精确的一个网络可靠性指标.一个图是Superλ-′的,如果它的任一最小限制边割都孤立一条有最小边度的边.本文考虑一类重要的网络模型-无向K autz图UK(d,n)的限制边连通度λ,′证明了当d 3,n 2时,λ(′UK(d,n))=4d-4,并进一步指出此时的UK(d,n)是Superλ-′的.     

Feedback numbers of Kautz digraphs

A subset of vertices (resp. arcs) of a graph G is called a feedback vertex (resp. arc) set of G if its removal results in an acyclic subgraph. Let f(d,n) (f_a(d,n)) denote the minimum cardinality over all feedback vertex (resp. arc) sets of the Kautz digraph K(d,n). This paper proves that for any integers d?2 and n?1

     

固定合作规模网络的平均路径长度

本文提出了一种特殊的合作网络,称之为固定合作规模网络．我们重点研究了这类网络的平均路径长度,通过建立微分方程,得到平均路径长度的增加速度近似与网络规模的对数成正比．     

The Independence Number for De Bruijn networks and Kautz networks

Let D be a directed graph; the (l,ω)-Independence Number of graph D, denoted by α_l,ω(D), is an important performance parameter for interconnection networks. De Bruijn networks and Kautz networks, denoted by B(d,n) and K(d,n) respectively, are versatile and efficient topological structures of interconnection networks. For l=1,2,…,n, this paper shows that α_l,d−1(B(d,n))=dⁿ,α_l,d−1(K(d,n))=α_l,d(K(d,n))=dⁿ+dⁿ⁻¹ if d≥3 and n≤d−2. In particular, the paper shows the exact value of the Independence Number for B(d,1) and B(d,2) for any d. For the generalized situation, the paper obtains a lower bound α_l,d−1(B(d,n))≥d² if n≥3 and d≥5.     

Bisecting de Bruijn and Kautz graphs

De Bruijn and Kautz graphs have been intensively studied as perspective interconnection networks of massively parallel computers. One of the crucial parameters of an interconnection network is its bisection width. It has an influence on both communication properties of the network and the algorithmic design. We prove optimal bounds on the edge and vertex bisection widths of the k-ary n-dimensional de Bruijn digraph. This generalizes known results for k = 2 and improves the upper bound for the vertex bisection width. We extend the method to prove optimal upper and lower bounds on the edge and vertex bisection widths of Kautz graphs.     

Kautz图的限制边连通度

限制边连通度是对传统边连通度的推广 ,而且是计算机互连网络容错性的一个重要度量 .本文考虑两类重要的网络模型———Kautz有向图K(d ,n)和Kautz无向图UK(d ,n)的限制边连通度λ′,并得到如下结果 :除了λ′(K( 2 ,1) )不存在外 ,均有λ′(K(d ,n) ) =2d-2 ;当d≥ 3 ,n≥ 3时 ,4d-5≤λ′(UK(d ,n) ) ≤ 4d -4 .     

Virtual Path Design in Service-Specific ATM Networks

This paper addresses the problem of virtual path management in ATM networks, which is the problem of jointly selecting efficient virtual trunk routes and sizing them to meet end-to-end grade-of-service requirements. The problem is posed over capacitated networks and is formulated as a two-level multi-commodity network flow problem with linear side-constraints (physical layer capacity) and non-linear side constraints (end-to-end/link blocking). Through a variety of examples we show the method (i) generates solutions that agree with engineering judgement, (ii) can solve VP layout management for realistic size networks (of up to 200 nodes) in reasonable time and (iii) provides upper bounds on how far the solution strays from the mathematically optimal design.     

Distance domination of generalized de Bruijn and Kautz digraphs

Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ _k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G _B (n, d) and generalized Kautz digraphs G _K (n, d) are good candidates for interconnection networks. Denote Δ _k := (∑ _j=0 ^k d ^j )^?1. F. Tian and J. Xu showed that ?nΔ _k ? γ _k (G _B (n, d)) ≤?n/d ^k? and ?nΔ _k ? ≤ γ _k (G _K (n, d)) ≤ ?n/d ^k ?. In this paper, we prove that every generalized de Bruijn digraph G _B (n, d) has the distance k-domination number ?nΔ _k ? or ?nΔ _k ?+1, and the distance k-domination number of every generalized Kautz digraph G _K (n, d) bounded above by ?n/(d ^k?1+d ^k )?. Additionally, we present various sufficient conditions for γ _k (G _B (n, d)) = ?nΔ _k ? and γ _k (G _K (n, d)) = ?nΔ _k ?.