完全对换网络的限制连通度

完全对换网络是基于 Cayley 图模型的一类重要互连网络. 一个图 G 的 k-限制点(边)连通度是使得 G-F 不连通且每个分支至少有 k 个顶点的最小点(边)子集 F 的基数, 记作 \kappa_{k}(\lambda_{k}). 它是衡量网络可靠性的重要参数之一, 也是图的容错性的一种精化了的度量. 一般地, 网络的 k-限制点(边)连通度越大, 它的连通性就越好. 证明了完全对换网络 CT_{n} 的 2-限制点(边)连通度和 3-限制点(边)连通度, 具体来说: 当 n\geq4 时, \kappa_{2}(CT_{n})=n(n-1)-2, \kappa_{3}(CT_{n})=\frac{3n(n-1)}{2}-6; 当 n\geq3 时, \lambda_{2}(CT_{n})=n(n-1)-2, \lambda_{3}(CT_{n})=\frac{3n(n-1)}{2}-4.     

关于Mbius立方体网格的连通度(英文)

图的连通度、超连通性和限制连通度是度量互连网络容错性的重要参数 .该文考虑n维M bius立方体网络MQn,证明了它的点和边连通度都为n ,当n是任何正整数时它是超连通的 ,当n≠ 2时它是超边连通的 ,当n≥ 3时它的限制点连通度和当n≥ 2时的限制边连通度都为 2n- 2 .     

笛卡尔乘积图的圈点连通度

图G的圈点连通度,记为κ_c(G),是所有圈点割中最小的数目,其中每个圈点割S满足G-S不连通且至少它的两个分支含圈.这篇文章中给出了两个连通图的笛卡尔乘积的圈点连通度:(1)如果G_1≌K_m且G_2≌K_n,则κ_c(G_1×G_2)=min{3m+n-6,m+3n-6},其中m+n≥8,m≥n+2,或n≥m+2,且κ_c(G_1×G_2)=2m+2n-8,其中m+n≥8,m=n,或n=m+1,或m=n+11;(2)如果G_1≌K_m(m≥3)且G_2■K_n,则min{3m+κ(G_2)-4,m+3κ(G_2)-3,2m+2κ(G_2)-4}≤κ_c(G_1×G_2)≤mκ(G2);(3)如果G_1■K_m,K_(1,m-1)且G_2■K_n,K_(1,n-1),其中m≥4,n≥4,则min{3κ(G_1)+κ(G_2)-1,κ(G_1)+3κ(G_2)-1,2_κ(G_1)+2_κ(G_2)-2}≤κ_c(G_1×G_2)≤min{mκ(G_2),nκ(G_1),2m+2n-8}.     

超级限制边连通二部图的充分条件

设S是连通图G的一个边割.若G-S不包含孤立点,则称S是G的一个限制边割.图G的最小限制边割的边数称为G的限制边连通度,记为λ'(G).如果图G的限制边连通度等于其最小边度,则称图G是最优限制边连通的,简称λ'-最优的.进一步,如果图G的每个最小限制边割恰好分离出图G的一条边,则称图G是超级限制边连通的,简称超级-λ'的.设G是一个最小度δ(G)≥2的n≥4阶二部图,ξ(G)是G的最小边度.本文证明了(a)若ξ(G)≥(n/2-2)(1+1/δ(G)-1),则G是λ'-最优的;(b)若ξ(G)>(n/2-2)(1+1/δ(G)-1),则G是超级-λ'的,除非图G是K2,n-2,n≥6或是Cartesian积图Kn/4,n/4×K2,其中n≥8且n整除4.最后,论文举例说明该结果是最好可能的.     

两类网络的2-限制连通度

给定图G=(V,E)和非负整数h,图G的h-限制点割S是V(G)的一个子集(如果存在)使得G-S不连通且G-S中任一点的度数至少为h.图G的h-限制连通度κ~h(G)是G的最小h-限制点割的阶数.本文中,我们证明了κ~2(FCQ_n)=4n-4 (n≥8),κ~2(SQn)=4n-8(n≥4),其中FCQ_n和SQ_n分别是n维折叠交叉超立方体和n维spined cube.     

折叠交叉立方体的分支边连通度

r-分支连通度(边连通度)是衡量大型互连网络可靠性和容错性的一个重要参数.设G是连通图且r是非负整数,如果G中存在某种点子集(边子集)使得G删除这种点子集(边子集)后得到的图至少有r个连通分支.则所有这种点子集(边子集)中基数最小的点子集(边子集)的基数称为图G的r-分支连通度(边连通度).n-维折叠交叉立方体FCQn是由交叉立方体CQn增加2n-1条边后所得.该文利用r-分支边连通度作为可靠性的重要度量,对折叠交叉立方体网络的可靠性进行分析,得到了折叠交叉立方体网络的2-分支边连通度,3-分支边连通度,4分支边连通度.确定了折叠交叉立方体FCQn的r-分支边连通度.     

单圈图的代数连通度的排序

n阶图G称为是一个单圈图,如果G是连通的,并且G的边数也是n.用U(n)表示所有n阶单圈图所成的集合.给出了当阶数n≥25时,代数连通度为前九大的n阶单圈图及它们的代数连通度.     

随机图的广义3-连通度

图的广义连通度的概念是由Chartrand等人引入的.令S表示图G的一个非空顶点集,κ(S)表示图G中连结S的内部不交树的最大数目.那么,对任意一个满足2≤r≤n的整数r,定义G的广义r-连通度为所有κ(S)中的最小值,其中S取遍G的顶点集合的r-元子集.显然,κ_2(G)=κ(G),即为图G的顶点连通度.所以广义连通度是经典连通度的一个自然推广.本文研究了随机图的广义3-连通度,证明了对任一给定的整数k,k≥1,p=(log n+(k+1)log long n-log lon logn)/n是关于性质κ_3(G(n,p))≥k的紧阈值函数.我们得到的结果可以看作是Bollobas和Thomason给出的关于经典连通度结果的推广.     

复合图及其连通度和临界度

本文确定了H(G)的点-连通度及H(G)关于点-连通度的临界度;确确定了H(G)的边-连通度及某种H(G)关于边-连通度的临界度。     

强哈密尔顿连通有向图的一个注记

利用收缩技术,证明了1)阶为n=2k且最小半度至少是k的有向图D是强哈密尔顿连通的,除非D属于某些图类;2)2强连通且包含n个顶点、(n-1)(n-2)+4条弧的有向图是强哈密尔顿连通的,除非D属于某些图类.     

On Eccentric Connectivity Index and Connectivity

Let G be a finite connected graph. The eccentric connectivity index ξ^c(G) of G is defined as ξ^c(G)= _{v∈V (G)} ec(v)deg(v), where ec(v) and deg(v) denote the eccentricity and degree of a vertex v in G, respectively. In this paper, we give an asymptotically sharp upper bound on the eccentric connectivity index in terms of order and vertex-connectivity and in terms of order and edge-connectivity. We also improve the bounds for triangle-free graphs.     

Connectivity interdiction

We consider the problem of maximally decreasing the edge-connectivity of an edge-weighted graph by removing a limited set of edges. This problem, which we term connectivity interdiction, falls into a large family of so-called interdiction problems, which have been considered in a variety of contexts. Whereas little is known about the approximability of most interdiction problems, we show that connectivity interdiction admits a PTAS, and a natural special case of it can even be solved efficiently.     

Connectivity calculus

Given a finite hypergraph H = (V, E) and, for each e ϵ E, a collection of nonempty subsets π_e of e, Möbius inversion is used to establish a recursive formula for the number of connected components of the hypergraph H = (V, ∪_eϵEπ_e). As shown elsewhere, this formula is an essential ingredient in the context of a certain divide-and-conquer strategy that allows us to define a dynamical programming scheme solving Steiner's problem for graphs in linear time (however, with a constant depending hyperexponentially on their tree width).     

On Vertex Connectivity and Absolute Algebraic Connectivity for Graphs

Let G be a graph on n vertices with vertex connectivity v with 1 h v h n m 2. We produce an attainable upper bound on the absolute algebraic connectivity of G in terms of n and v .     

On Vertex Connectivity and Absolute Algebraic Connectivity for Graphs

Let G be a graph on n vertices with vertex connectivity v with 1 ≤ v ≤ n -2. We produce an attainable upper bound on the absolute algebraic connectivity of G in terms of n and v .     

Decremental Dynamic Connectivity

We consider Las Vegas randomized dynamic algorithms for on-line connectivity problems with deletions only. In particular, we show that starting from a graph with m edges and n nodes, we can maintain a spanning forest during m deletions in O(m log(n²/m) + n(log n)³(log log n)²) expected time, which is O(m) if m = Θ(n²) and O(m log n) if m = Ω(n(log n log log n)²). The deletions may be interspersed with connectivity queries, each of which is answered in constant time. The previous best bound was O(m log² n) by Henzinger and Thorup which covered both insertions and deletions. The result is based on a general randomized reduction for edge connectivity problems of many deletions-only queries to a few deletions and insertions queries. For 2-edge connectivity, the complexity is improved from O(m(log n)⁵) to O(m log(n²/m) + n(log n)⁶(log log n)²). For the general decremental k-edge-connectivity problem, we get a total running time of O(k²n² polylog n). Here the previous best bound was O(kmn polylog n). Improved running times are also achieved for the static consensus tree problem, with applications to computational biology and relational data bases.     

Flexible Graph Connectivity

Mathematical Programming - We introduce and study the problem Flexible Graph Connectivity, which in contrast to many classical connectivity problems features a non-uniform failure model. We...     

Connectivity in digraphs

Let c(x,y) denote the maximum number of edge-disjoint directed paths joining x to y in the digraph G. It is shown that, for a given point a of G, c(a,x) ≤ c(x,a) for any x implies that the outdegree of a is ≤ its indegree. An immediate consequence is Kotzig's conjecture: Given a digraph G, c(x,y) = c(y,x) for every x, y if and only if the graph is pseudo-symmetric, i.e., each point has the same indegree and outdegree (the “if” part having been proved by Kotzig). The same method is applied to prove a weakened form of a conjecture of N. Robertson, while the original conjecture is disproved.     

Super Connectivity and Super Edge Connectivity of the Mycielskian of a Graph

Mycielski introduced a new graph transformation μ(G) for graph G, which is called the Mycielskian of G. A graph G is super connected or simply super-κ (resp. super edge connected or super-λ), if every minimum vertex cut (resp. minimum edge cut) isolates a vertex of G. In this paper, we show that for a connected graph G with |V(G)| ≥ 2, μ(G) is super-κ if and only if δ(G) < 2κ(G), and μ(G) is super-λ if and only if G\ncong K₂{G\ncong K_2}.