周长为4不含3圈的m限制边连通图

m限制边割是连通图的一个边割,它将此图分离成阶不小于m的连通分支刻画了周长为4,不含3圈的m限制边割的图类.     

图的3限制性边割

3限制性边割将连通图分离成不连通图，使其各连通分支含有至少3个顶点．含3限制性边割的图在本文中得到刻划．     

有向图的边割(X,Y)中｜X｜和｜Y｜的下界与有向图的极大性和超级性

在已有的极大边连通、超级边连通、极大局部边连通有向图概念的基础上,提出超级局部边连通有向图的概念,对一般的、二部的、基础图的团数至多为p的有向图、定向图分别给出｜(X,Y)｜＜δ(D)的边割(X,Y)、非平凡的最小边割(X,Y)中｜X｜和｜Y｜的下界,据此分别得到极大边连通、超级边连通有向图的最小度条件.类似地分别得到...     

3连通图的可去边数

设 e是 3连通图 G的一条边 ,如果 G- e是某个 3连通图的剖分 ,则称 e是 G的可去边 .本文给出了 3连通图的可去边数依赖于极大半轮的下界以及达到下界的极图 .     

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 .     

4连通图的可去边与4连通图的构造

本文引进了４连通图的可去边的概念,,并证明了４连通图Ｇ中不存在可去边的充要条件是Ｇ＝Ｃ５或Ｃ６,同时给出了ｎ阶４连通图的一个新的构造方法．     

周长为3的m限制边割连通图

如果连通图的G存在边割S,使得G-S的每一个连通分支都含有至少m个顶点,则称图G是m限制边连通的.本文刻画了周长为3的m限制边连通图.     

极大3限制边连通二部图的充分条件

设G=(V,E)是一个连通图.称一个边集合S■E是一个k限制边割,如果G-S的每个连通分支至少有k个顶点.称G的所有k限制边割中所含边数最少的边割的基数为G的k限制边连通度,记为λ_k(G).定义ξ_k(G)=min{[X,■]:|X|=k,G[X]连通,■=V(G)\X}.称图G是极大k限制边连通的,如果λ_k(G)=ξ_k(G).本文给出了围长为g>6的极大3限制边连通二部图的充分条件.     

恰有k条非基本边的极小3连通图

设G是简单3连通图．G\e(删除边e)和G/e(收缩边e)都不是简单3连通图,则e称为G的基本边．对于3连通图中的非基本边．Tutte证明了：唯一没有非基本边的简单3连通图是轮．Oxley和Wu确定了至多有3条非基本边的所有极小3连通图以及恰有4条非基本的极小3连通图．Reid与Wu确定了至多有5条非基本边的极小3连通图．在本文中,我们在极小3连通图中定义了三种运算,然后通过轮利用这些运算的逆运算给出恰有k(k■2)条非基本边的极小3连通图的一种构造方法．     

3-连通3-正则图生成树外的可去边

G是3-连通图，e是G中的一条边．若G-e是3-连通图的一个剖分，则称e是3-连通图的可去边．否则，e是G中不可去边．本给出3-连通3-正则图中生成树外可去边的分布情况及数目．     

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.     

Super cyclically edge-connected vertex-transitive graphs of girth at least 5

A cyclic edge-cut of a graph G is an edge set, the removal of which separates two cycles. If G has a cyclic edge-cut, then it is called cyclically separable. We call a cyclically separable graph super cyclically edge-connected, in short, super-λc, if the removal of any minimum cyclic edge-cut results in a component which is a shortest cycle. In [Zhang, Z., Wang, B.: Super cyclically edge-connected transitive graphs. J. Combin. Optim., 22, 549-562 (2011)], it is proved that a connected vertex-transitive graph is super-λc if G has minimum degree at least 4 and girth at least 6, and the authors also presented a class of nonsuper-λc graphs which have degree 4 and girth 5. In this paper, a characterization of k (k≥4)-regular vertex-transitive nonsuper-λc graphs of girth 5 is given. Using this, we classify all k (k≥4)-regular nonsuper-λc Cayley graphs of girth 5, and construct the first infinite family of nonsuper-λc vertex-transitive non-Cayley graphs.     

Sufficient Conditions for Maximally Edge-connected and Super-edge-connected Digraphs Depending on the Size

Let D be a finite and simple digraph with vertex set V (D). The minimum degree δ of a digraph D is defined as the minimum value of its out-degrees and its in-degrees. If D is a digraph with minimum degree δ and edge-connectivity λ, then λ ≤ δ. A digraph is maximally edge-connected if λ=δ. A digraph is called super-edge-connected if every minimum edge-cut consists of edges incident to or from a vertex of minimum degree. In this note we show that a digraph is maximally edge-connected or super-edge-connected if the number of arcs is large enough.     

图的笛卡儿积及字典式积的连通性

文章给出了两个图的笛卡儿积及字典式的积为最大边连通的、最大连通的、super-λ,super-κ及hyper-κ的充分条件,同时证明了其中一些条件也是必要的.此外,对这两种积的局部割集和广义割集的性质也进行了考虑.     

SUPER EDGE-CONNECTIVITY OF DE BRUIJN AND KAUTZ UNDIRECTED GRAPHS

The super edge-connectivity of a graph is an important parameter to measure fault-tolerance of interconnection networks. This note shows that the Kautz undirected graph is super edge-connected,and provides a short proof of Lue and Zhang‘s result on super edge-connectivity of the de Bruijn undirected graph.     

极大限制边连通超图的两个充分条件

图的限制边连通度是经典边连通度的推广,可用于精确度量网络的容错性.极大限制边连通图是使限制边连通度达到最优的一类图.首先将图的限制边连通度和最小边度的概念推广到r一致线性超图H,证明当H的最小度δ(H)≥r+1时,H的最小边度ξ(H)是它的限制边连通度λ′(H)的一个上界,并将满足ξ(H)=λ′(H)的H称为极大限制边连通超图,然后证明n个顶点的r一致线性超图H如果满足δ(H)≥(n-1)/(2(r-1))+(r-1),则它是极大限制边连通的,最后证明直径为2,围长至少为4的一致线性超图是极大限制边连通的.所得结论是图中相关结果的推广.     

Super Edge-connectivity and Zeroth-order General Randić Index for −1 ≤ <Emphasis Type="Italic">α</Emphasis> &lt; 0

Let G be a connected graph with order n, minimum degree δ = δ(G) and edge-connectivity λ = λ(G). A graph G is maximally edge-connected if λ = δ, and super edge-connected if every minimum edgecut consists of edges incident with a vertex of minimum degree. Define the zeroth-order general Randi? index \(R_\alpha ^0\left( G \right) = \sum\limits_{x \in V\left( G \right)} {d_G^\alpha \left( x \right)} \), where d_G(x) denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super edge-connected in terms of the zeroth-order general Randi? index for ?1 ≤ α < 0, respectively.     

Maximally edge-connected and vertex-connected graphs and digraphs: A survey

Let D be a graph or a digraph. If δ(D) is the minimum degree, λ(D) the edge-connectivity and κ(D) the vertex-connectivity, then κ(D)?λ(D)?δ(D) is a well-known basic relationship between these parameters. The graph or digraph D is called maximally edge-connected if λ(D)=δ(D) and maximally vertex-connected if κ(D)=δ(D). In this survey we mainly present sufficient conditions for graphs and digraphs to be maximally edge-connected as well as maximally vertex-connected. We also discuss the concept of conditional or restricted edge-connectivity and vertex-connectivity, respectively.     

图的超级限制边连通性

在Moor-Shannon网络模型中,边连通度和限制边连通度较大的网络一般有较好的可靠性和容错性.本文证明:除两种平凡情形外,无向Kautz网络的拓扑结构,无向Kautz图UK(2,n)是超级限制边连通的.因此,它们比de Bruijn网络有更好的限制边连通性.     

最大边连通和super-边连通超图的充分条件

设H是连通超图。若超图H的边连通度等于其最小度,则称H是最大边连通的。若超图H的每个最小边割总是由关联于某个最小度顶点的边集所构成,则称H是super-边连通的。首先给出一致线性超图是最大边连通超图的度序列条件。其次,给出一致线性超图是super-边连通超图的度条件。这些结果分别推广了Dankelmann和Volkmann（1997）以及Hellwig和Volkmann（2005）在图上的相关结论。