On super-edge-connected digraphs and bipartite digraphs

A maximally edge-connected digraph is called super-λ if every minimum edge disconnecting set is trivial, i.e., it consists of the edges adjacent to or from a given vertex. In this paper sufficient conditions for a digraph to be super-λ are presented in terms of parameters such as diameter and minimum degree. Similar results are also given for bipartite digraphs. As a corollary, some characterizations of super-λ graphs and bipartite graphs are obtained. © 1929 John Wiley & Sons, Inc.     

Maximally connected digraphs

This paper introduces a new parameter I = I(G) for a loopless digraph G, which can be thought of as a generalization of the girth of a graph. Let k, λ, δ, and D denote respectively the connectivity, arc-connectivity, minimum degree, and diameter of G. Then it is proved that λ = δ if D ? 2I and κ k = δ if D ? 2I - 1. Analogous results involving upper bounds for k and λ are given for the more general class of digraphs with loops. Sufficient conditions for a digraph to be super-λ and super-k are also given. As a corollary, maximally connected and superconnected iterated line digraphs and (undirected) graphs are characterized.     

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.     

有向图的L(j,k)数的上界

给定正整数j≥k,有向图D的一个L(j,k)-标号是指从V(D)到非负整数集的一个函数f,使得当x在D中邻接到y时|f(x)-f(y)|≥j,当x在D中到y距离为二时|f(x)-f(y)|≥k.f的像元素称为标号.L(j,k)一标号问题就是确定(?)j,k-数(?)j,k(D),这个参数等于(?) max{f(x)|x∈V(D)},这里f取遍D的所有L(j,k)-标号.本文根据有向图的有向着色数及最长有向路的长度来研究(?)j,k-数,证明了:(1)对任何有向着色数为(?)(D)的有向图D,(?)j,k(D)≤((?)(D)-1)j;(2)对任何最长有向路的长度为l的有向图D,如果不含有向圈或者D中最长有向圈长度为l 1,则(?)j,k(D)≤lj.并且这两个界都是可达的.最后我们对l=3的有向图给出了3j-L(j,k)-labelling的一个有效算法.     

完备矩阵代数Ⅰ——乘法连续性问题

<正> 完备空间与完备矩阵环是Kothe.G.于1934年引入的,近年来由于在分析中最有用的一类线性拓扑空间——核空间的种种研究的影响,又给Kothe理论带来了许多新的进展,出现了大量文章.然而,至今为止还未见到有把完备矩阵环当作拓扑代数来加以探讨的工作,为此,我们准备着手对这样一类显然有其重要性的拓扑代数进行详细的讨论.     

Domination Number of Graphs Without Small Cycles

It has been shown (J. Harant and D. Rautenbach, Domination in bipartite graphs. Discrete Math. 309:113–122, 2009) that the domination number of a graph of order n and minimum degree at least 2 that does not contain cycles of length 4, 5, 7, 10 nor 13 is at most \frac3n8{\frac{3n}{8}}. They believed that the assumption that the graphs do not contain cycles of length 10 might be replaced by the exclusion of finitely many exceptional graphs. In this paper, we positively answer that if G is a connected graph of order n and minimum degree at least 2 that does not contain cycles of length 4, 5 nor 7 and is not one of three exceptional graphs, then the domination number of G is at most \frac3n8{\frac{3n}{8}}.     

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.     

有向线图的限制性连通度

设D=（y（D）,A（D））是一个强连通有向图．弧集S A（D）称为D的k-限制性弧割,如果D-S中至少有两个强连通分支的阶数大于等于后．最小k-限制性弧割的基数称为k-限制性弧连通度,记作Ak（D）．k-限制性点连通度Kk（D）可以类似地定义．有k-限制性弧割（k-限制性点割）的有向图称为λk-连通（kk-连通）有向图．本文研究有向图D的限制性弧连通度和其线图L（D）的限制性点连通度的关系,证明了对任意λk-连通有向图D,kk（L（D））≤λk（D）,当k=2,3时等式成立;若L（D）是Kk（k-1）连通的,则λk（D）≤Kk（k-1）（L（D））;特别地,若D是一个定向图且L（D）是Kk（k-1）／2．连通的,贝0Ak（D）≤Kk（k-1）,2（L（D））．     

The connectivity of large digraphs and graphs

This paper studies the relation between the connectivity and other parameters of a digraph (or graph), namely its order n, minimum degree δ, maximum degree Δ, diameter D, and a new parameter l_pi;, 0 ≤ π ≤ δ ? 2, related with the number of short paths (in the case of graphs l₀ = ?(g ? 1)/2? where g stands for the girth). For instance, let G = (V,A) be a digraph on n vertices with maximum degree Δ and diameter D, so that n ≤ n(Δ, D) = 1 + Δ + Δ ² + … + Δ^D (Moore bound). As the main results it is shown that, if κ and λ denote respectively the connectivity and arc-connectivity of G, . Analogous results hold for graphs. © 1993 John Wiley & Sons, Inc.     

Edge-connectivity in p-partite graphs

Let G = (V, E) be a finite, simple p-partite graph with minimum degree δ and edge-connectivity γ. It is proved that if |V| ? (2pδ)/(p - 1) - 2 or in special cases that if |V| ? (2pδ)/(p - 1) - 1, then λ = δ. It is further shown that this result is best possible.     

关于q-树二次整子图和n阶加点q-树色多项式的注记

在这篇文章中我们成功地仅用色多项式表征了最小度不等于q-3的q-树的二次整子图和n阶加点q-树,即当图的最小度δ(G)≠q-3时,n阶图G具有色多项式P(G;λ)=λ(λ-1)…(λ-q+2)(λ-q+1)~3(λ-q)~(n-q-2), n≥q+2,当且仅当G是n阶q-树的二次整子图或n阶加点q-树.     

具有最小弧数且基指数为3的本原不可幂对称无环带号有向图

Let S be a primitive non-powerful symmetric loop-free signed digraph on even n vertices with base 3 and minimum number of arcs. In [Lihua YOU, Yuhan WU. Primitive non-powerful symmetric loop-free signed digraphs with given base and minimum number of arcs. Linear Algebra Appl., 2011, 434(5), 1215-1227], authors conjectured that D is the underlying digraph of S with exp(D) = 3 if and only if D is isomorphic to ED n,3,3 , where ED n,3,3 = (V, A) is a digraph with V = {1, 2, . . . , n}, A = {(1, i), (i, 1) | 3≤i≤n} ∪ {(2i-1, 2i), (2i, 2i-1) | 2≤i≤ n/2 } ∪ {(2, 3), (3, 2), (2, 4), (4, 2)}). In this paper, we show the conjecture is true and completely characterize the underlying digraphs which have base 3 and the minimum number of arcs.     

Sufficient conditions for graphs to be λ′‐optimal,super‐edge‐connected,and maximally edge‐connected

The restricted‐edge‐connectivity of a graph G, denoted by λ′(G), is defined as the minimum cardinality over all edge‐cuts S of G, where G‐S contains no isolated vertices. The graph G is called λ′‐optimal, if λ′(G) = ξ(G), where ξ(G) is the minimum edge‐degree in G. A graph is super‐edge‐connected, if every minimum edge‐cut consists of edges adjacent to a vertex of minimum degree. In this paper, we present sufficient conditions for arbitrary, triangle‐free, and bipartite graphs to be λ′‐optimal, as well as conditions depending on the clique number. These conditions imply super‐edge‐connectivity, if δ (G) ≥ 3, and the equality of edge‐connectivity and minimum degree. Different examples will show that these conditions are best possible and independent of other results in this area. © 2005 Wiley Periodicals, Inc. J Graph Theory 48: 228–246, 2005     

A note on 3-blocked designs

A blocking set of a design different from a 2-(λ + 2, λ + 1, λ) design has at least 3 points. The aim of this note is to establish which 2-(v, k, λ) designs D with r ≥ 2λ may contain a blocking 3-set. The main results are the following. If D contains a blocking 3-set, then D is one of the following designs: a 2-(2λ + 3, λ + 1, λ), a 2-(2λ + 1), λ + 1, λ), a 2-(2λ - 1, λ, λ), a 2-(4λ + 3, 2λ + 1, λ) Hadamard design with λ odd, or a 2-(4λ - 1, 2λ, λ) Hadamard design. Moreover a blocking 3-set in a 2-(4λ + 3, 2λ + 1, λ) Hadamard design exists if and only if there is a line with three points. In the case of 2- (4λ - 1, 2λ, λ) Hadamard design with λ odd, we give necessary and sufficient conditions for the existence of a blocking 3-set, while in the case λ even, a necessary condition is given. © 1997 John Wiley & Sons, Inc.     

Approximation Schemes for the Generalized Traveling Salesman Problem

A graph Γ is called a Deza graph if it is regular and the number of common neighbors of any two distinct vertices is one of two fixed values. A Deza graph is called a strictly Deza graph if it has diameter 2 and is not strongly regular. In 1992, Gardiner et al. proved that a strongly regular graph that contains a vertex with disconnected second neighborhood is a complete multipartite graph with parts of the same size greater than 2. In this paper, we study strictly Deza graphs with disconnected second neighborhoods of vertices. In Section 2, we prove that, if each vertex of a strictly Deza graph has disconnected second neighborhood, then the graph is either edge-regular or coedge-regular. In Sections 3 and 4, we consider strictly Deza graphs that contain at least one vertex with disconnected second neighborhood. In Section 3, we show that, if such a graph is edge-regular, then it is the s-coclique extension of a strongly regular graph with parameters (n, k, λ, μ), where s is an integer, s ≥ 2, and λ = μ. In Section 4, we show that, if such a graph is coedge-regular, then it is the 2-clique extension of a complete multipartite graph with parts of the same size greater than or equal to 3.     

Slow mixing of Glauber dynamics for the hard‐core model on regular bipartite graphs

Let ∑ = (V,E) be a finite, d‐regular bipartite graph. For any λ > 0 let π_λ be the probability measure on the independent sets of ∑ in which the set I is chosen with probability proportional to λ^|I| (π_λ is the hard‐core measure with activity λ on ∑). We study the Glauber dynamics, or single‐site update Markov chain, whose stationary distribution is π_λ. We show that when λ is large enough (as a function of d and the expansion of subsets of single‐parity of V) then the convergence to stationarity is exponentially slow in |V(∑)|. In particular, if ∑ is the d‐dimensional hypercube {0,1}^d we show that for values of λ tending to 0 as d grows, the convergence to stationarity is exponentially slow in the volume of the cube. The proof combines a conductance argument with combinatorial enumeration methods. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

On Moore bipartite digraphs

In the context of the degree/diameter problem for directed graphs, it is known that the number of vertices of a strongly connected bipartite digraph satisfies a Moore‐like bound in terms of its diameter k and the maximum out‐degrees (d₁, d₂) of its partite sets of vertices. It has been proved that, when d₁d₂ > 1, the digraphs attaining such a bound, called Moore bipartite digraphs, only exist when 2 ≤ k ≤ 4. This paper deals with the problem of their enumeration. In this context, using the theory of circulant matrices and the so‐called De Bruijn near‐factorizations of cyclic groups, we present some new constructions of Moore bipartite digraphs of diameter three and composite out‐degrees. By applying the iterated line digraph technique, such constructions also provide new families of dense bipartite digraphs with arbitrary diameter. Moreover, we show that the line digraph structure is inherent in any Moore bipartite digraph G of diameter k = 4, which means that G = L G′, where G′ is a Moore bipartite digraph of diameter k = 3. © 2003 Wiley Periodicals, Inc. J Graph Theory 43: 171–187, 2003     

Supereulerian bipartite digraphs

A digraph D is supereulerian if D has a spanning closed ditrail. Bang‐Jensen and Thomassé conjectured that if the arc‐strong connectivity of a digraph D is not less than the independence number , then D is supereulerian. A digraph is bipartite if its underlying graph is bipartite. Let be the size of a maximum matching of D. We prove that if D is a bipartite digraph satisfying , then D is supereulerian. Consequently, every bipartite digraph D satisfying is supereulerian. The bound of our main result is best possible.     

Double-super-connected digraphs

A strongly connected digraph D is said to be super-connected if every minimum vertex-cut is the out-neighbor or in-neighbor set of a vertex. A strongly connected digraph D is said to be double-super-connected if every minimum vertex-cut is both the out-neighbor set of a vertex and the in-neighbor set of a vertex. In this paper, we characterize the double-super-connected line digraphs, Cartesian product and lexicographic product of two digraphs. Furthermore, we study double-super-connected Abelian Cayley digraphs and illustrate that there exist double-super-connected digraphs for any given order and minimum degree.     

Nonlinear eigenvalue problems with positively convex operators

We consider the equation u = λAu (λ > 0), where A is a forced isotone positively convex operator in a partially ordered normed space with a complete positive cone K. Let Λ be the set of positive λ for which the equation has a solution u?K, and let Λ₀ be the set of positive λ for which a positive solution—necessarily the minimum one—can be obtained by an iteration u_n = λAu_n?1, u₀ = 0. We show that if K is normal, and if Λ is nonempty, then Λ₀ is nonempty, and each set Λ₀, Λ is an interval with $\text{inf}\text{(Λ}{}_0\text{) =}\text{inf}\text{(Λ) = 0}\text{and sup}\text{(Λ}{}_0\text{) =}\text{sup}\text{(Λ) (= λ1,}\text{say}\text{)}$; but we may have λ1 ? Λ₀ and λ1 ? Λ. Furthermore, if A is bounded on the intersection of K with a neighborhood of 0, then Λ₀ is nonempty. Let u⁰(λ) = lim_n→∞(λA)ⁿ(0) be the minimum positive fixed point corresponding to λ ? Λ₀. Then u⁰(λ) is a continuous isotone convex function of λ on Λ₀.