Clique Numbers of Random Subgraphs of Some Distance Graphs

We consider a class of graphs G(n, r, s) = (V (n, r),E(n, r, s)) defined as follows:

$$V(n,r) = \{ x = ({x_{1,}},{x_2}...{x_n}):{x_i} \in \{ 0,1\} ,{x_{1,}} + {x_2} + ... + {x_n} = r\} ,E(n,r,s) = \{ \{ x,y\} :(x,y) = s\} $$

where (x, y) is the Euclidean scalar product. We study random subgraphs G(G(n, r, s), p) with edges independently chosen from the set E(n, r, s) with probability p each. We find nontrivial lower and upper bounds on the clique number of such graphs.

     

On Stability of the Independence Number of a Certain Distance Graph

Problems of Information Transmission - We study the asymptotic behavior of the independence number of a random subgraph of a certain (r, s)-distance graph. We provide upper and lower...     

Graph Characterization by Counting Sink Star Subgraphs

In this paper, we study the polynomial coefficients of the reduced Bartholdi zeta function for characterizing simple unweighted graphs and demonstrate how to use these coefficients for clustering graphs. The polynomial coefficients of the reduced Bartholdi zeta function are invariant to vertex order permutations and also carry information about counting the sink star subgraphs in a symmetric digraph of G. We also investigate the advantages of the reduced Bartholdi coefficients over other spectral methods such as the Ihara zeta function and Laplacian spectra. Experimental results indicate that the proposed method is more effective than the other spectral approaches, and compared to the Ihara zeta function, it has less sensitivity to structural noises such as omitting an edge.     

从不确定图中发现K紧密子图

由蛋白质交互网络、社会网络及无线通信网络构成的图中存在许多不确定性。如何高效获取不确定图中有价值的信息,如蛋白质网络中关键的功能集团、社会网络中适于投放广告的团体及通信网络中应重点维护的区域等,具有重要的现实意义。从理论上证明了在不确定图中发现最紧密子图问题具有NP-Hard复杂性;基于树搜索策略提出了通过枚举解空间及剪枝获得最优解的算法TreeClose;针对树搜索算法TreeClose在处理大图时空间复杂度过高的问题,提出了基于贪心思想的2-近似算法GreedyClose。实验结果表明,通过上述算法可以高效快速地在不确定图中发现紧密子图,从而解决在实际应用中遇到的各种问题。     

用PB实现随机取数

本文介绍了PowerBuilder中的随机函数，以及随机函数中“种子”（Seed)的选取方法，并且给出了实现随机取数的算法框图和程序。     

图编辑距离概述

图编辑距离是图模式匹配技术中常用的方法之一。基于图编辑距离的匹配方法能够处理多种类型的图数据,因而受到了学术界的广泛关注。首先介绍了图编辑距离的相关概念;然后简述了基于启发式搜索技术的精确图编辑距离算法,重点分析了基于二分图匹配的近似图编辑距离算法;最后对现存的一些图编辑问题进行了总结,并对未来的发展趋势进行了展望。     

Verification of Random Graph Transformation Systems

In this paper we describe some statistical results obtained by the verification of random graph transformation systems (GTSs). As a verification technique we use over-approximation of GTSs by Petri nets. Properties we want to verify are given by markings of Petri nets. We also use counterexample-guided abstraction refinement approach to refine the obtained approximation. A software tool (Augur) supports the verification procedure. The idea of the paper is to see how many of the generated systems can be successfully verified using this technique.     

Planar Graph Coloring Avoiding Monochromatic Subgraphs: Trees and Paths Make It Difficult

We consider the problem of coloring a planar graph with the minimum number of colors so that each color class avoids one or more forbidden graphs as subgraphs. We perform a detailed study of the computational complexity of this problem. We present a complete picture for the case with a single forbidden connected (induced or noninduced) subgraph. The 2-coloring problem is NP-hard if the forbidden subgraph is a tree with at least two edges, and it is polynomially solvable in all other cases. The 3-coloring problem is NP-hard if the forbidden subgraph is a path with at least one edge, and it is polynomially solvable in all other cases. We also derive results for several forbidden sets of cycles. In particular, we prove that it is NP-complete to decide if a planar graph can be 2-colored so that no cycle of length at most 5 is monochromatic.     

On the Grundy and b-Chromatic Numbers of a Graph

The Grundy number of a graph G, denoted by Γ(G), is the largest k such that G has a greedy k-colouring, that is a colouring with k colours obtained by applying the greedy algorithm according to some ordering of the vertexes of G. The b-chromatic number of a graph G, denoted by χ _b (G), is the largest k such that G has a b-colouring with k colours, that is a colouring in which each colour class contains a b-vertex, a vertex with neighbours in all other colour classes. Trivially χ _b (G),Γ(G)≤Δ(G)+1. In this paper, we show that deciding if Γ(G)≤Δ(G) is NP-complete even for a bipartite graph G. We then show that deciding if Γ(G)≥|V(G)|?k or if χ _b (G)≥|V(G)|?k are fixed parameter tractable problems with respect to the parameter k.     

基于子图的随机图点覆盖2度点核化研究

点覆盖问题虽然可以在参数计算理论的架构内求精确解,但是目前在理论及应用上有一定的局限性.根据不同度的顶点之间及顶点与边的关系,提出随机图参数化点覆盖问题的d-核化可决策性及2度点三角形子图的计数方法;通过研究子图对顶点的共享关系,分析2度顶点核化过程中核及度分布演变的动态过程,得出随机图2度点核化强度与2度点概率关系及2度点核化可决策性的两个推论: 2度点核化算法对2度点分布概率约为0.75的随机图的核化强度最高;对顶点度概率分布为φ(x)的随机图的参数化点覆盖问题(G,k),当k小于某一与φ(x)有关的值时,它是2-核化可决策的.仿真结果证实,该理论能够把握2度点核化的内在机制,提供随机图上这一NP完全问题的求解方法,也为参数计算在已知度分布的一类不确定问题中的应用提供了可能.     

实现公共随机数同步的方法

实现公共随机数同步是仿真中获得输出估计量差衰减的关键，对随机变量分布函数的单调逼近能有效实现同步，并由此提出实现同步的方法。理论和实例显示所提方法行之有效的。     

General Independence Sets in Random Strongly Sparse Hypergraphs

We analyze the asymptotic behavior of the j-independence number of a random k-uniform hypergraph H(n, k, p) in the binomial model. We prove that in the strongly sparse case, i.e., where \(p = c/\left( \begin{gathered} n - 1 \hfill \\ k - 1 \hfill \\ \end{gathered} \right)\) for a positive constant 0 < c ≤ 1/(k ? 1), there exists a constant γ(k, j, c) > 0 such that the j-independence number α _j (H(n, k, p)) obeys the law of large numbers \(\frac{{{\alpha _j}\left( {H\left( {n,k,p} \right)} \right)}}{n}\xrightarrow{P}\gamma \left( {k,j,c} \right)asn \to + \infty \) Moreover, we explicitly present γ(k, j, c) as a function of a solution of some transcendental equation.     

求解连通无向简图的所有连通子图的算法及其应用

讨论了求解连通无向简图的所有连通子图的算法，并将其应用于化学领域：如何将许多具有某种共同属性的物质的分子结构图形分解成子分子结构，进一步地试图找出存在于大多数具有该属性的物质中的子分子结构，并讨论这样的子分子结构导致物质具有该共同属性的概率。     

关于Bubblesort-star网络的距离控制数

对于任意的正整数l,连通图G的顶点子集D被称为距离l-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u和v在G中的距离不超过l.图G的距离l-控制数是指G中所有距离l-控制集的最小基数,1-控制数常常称为控制数.给出了Bubblesort-star网络的控制数、距离2-控制数和距离3-控制数的界,而且针对某些低维Bubblesort-star网络的这几类控制数给出了更好的界.     

基于攻击距离的攻击图优化方法

传统网络攻击图的生成随着网络规模扩大存在状态爆炸问题,网络安全管理员往往拿着冗余的攻击图不知所措。为了消除攻击图中不必要的攻击路径,保留下最优的攻击路径以供管理员防御参考,本文利用攻击距离对复杂的攻击图进行了优化。实验结果表明,利用此方法优化后的攻击图保留了最有可能的攻击路径,降低了攻击图的规模,随着网络规模的扩大,效果也越来越明显。     

结点可同名的连通无向图的子图研究

给出了求解结点可同名的连通无向图的所有边极大连通子图的算法,并将其应用于化学领域: 将许多具有某种共同属性的物质的分子结构图形分解成子分子结构,进一步地试图找出存在于大多数具有该属性的物质中的子分子结构,并讨论这样的子分子结构导致物质具有该共同属性的可能性.     

结点可同名的连通无向图的子图研究

给出了求解结点可同名的连通无向图的所有边极大连通子图的算法,并将其应用于化学领域:将许多具有某种共同属性的物质的分子结构图形分解成子分子结构,进一步地试图找出存在于大多数具有该属性的物质中的子分子结构,并讨论这样的子分子结构导致物质具有该共同属性的可能性。     

基于环境噪声的真随机数的产生

为了提供一种简便和有效的真随机数生成方法,通过分析环境噪声数据的统计特性,提出同余法和正负法将声音波形数据进行处理。去除声波数据的周期性、连续性、相关性,提取其随机性。同真随机数采集系统设计相关的电路噪声问题以及采集速度问题也作了讨论。用此思路构建的真随机数采集系统原型已经能够用来产生性能较佳的真随机数,证明了设计思路和处理方法的正确有效性。     

Random Geometric Graph Diameter in the Unit Ball

The unit ball random geometric graph has as its vertices n points distributed independently and uniformly in the unit ball in , with two vertices adjacent if and only if their ℓ_p-distance is at most λ. Like its cousin the Erdos-Renyi random graph, G has a connectivity threshold: an asymptotic value for λ in terms of n, above which G is connected and below which G is disconnected. In the connected zone we determine upper and lower bounds for the graph diameter of G. Specifically, almost always, , where is the ℓ_p-diameter of the unit ball B. We employ a combination of methods from probabilistic combinatorics and stochastic geometry.