Finding the Maximal Eigenpair for a Large,Dense,Symmetric Matrix based on Mufa Chen's Algorithm

A hybrid method is presented for determining maximal eigenvalue and its eigenvector(called eigenpair)of a large,dense,symmetric matrix.Many problems require finding only a small part of the eigenpairs,and some require only the maximal one.In a series of papers,efficient algorithms have been developed by Mufa Chen for computing the maximal eigenpairs of tridiagonal matrices with positive off-diagonal elements.The key idea is to explicitly construet effective initial guess of the maximal eigenpair and then to employ a self-closed iterative algorithm.In this paper we will extend Mufa Chen's algorithm to find maximal eigenpair for a large scale,dense,symmetric matrix.Our strategy is to first convert the underlying matrix into the tridiagonal form by using similarity transformations.We then handle the cases that prevent us from applying Chen's algorithm directly,e.g.,the cases with zero or negative super-or sub-diagonal elements.Serval numerical experiments are carried out to demonstrate the efficiency of the proposed hybrid method.     

以子图为根的k距局部子图的可重构性

本文定义了图中以某个图为根的k距局部子图，证明了图中同构于上述 k距局部子图的子图的数目是可重构的，从而给出了一个新结果并推广了文献^[5]中的定理。     

On the Probability that a Random Subgraph Contains a Circuit

Let and . We show that, if G is a sufficiently large simple graph of average degree at least μ, and H is a random spanning subgraph of G formed by including each edge independently with probability , then H contains a cycle with probability at least .     

Subgraph Transversal of Graphs

Given graphs F and G, denote by \({\tau_F}(G)\) the cardinality of a smallest subset \(T {\subseteq}V(G)\) that meets every maximal F-free subgraph of G. Erdös, Gallai and Tuza [9] considered the question of bounding \(\tau_{\overline{K_2}}(G)\) by a constant fraction of |G|. In this paper, we will give a complete answer to the following question: for which F, is τ _F (G) bounded by a constant fraction of |G|?In addition, for those graphs F for which \({\tau_F}(G)\) is not bounded by any fraction of |G|, we prove that \(\tau_F(G)\le|G|-\frac{1}{2}\sqrt{|G|}+\frac{1}{2}\), provided F is not K _k or \(\overline{K_k}\).     

含有给定k-正则子图的[a,b]-因子

设G是一个图,并设n,k,r,a和b是整数且满足k≥1,k≤a＜b和n≥3.对于G的给定的k-正则图H,如果G是K1,n-free图,且G的最小度至少是((n(a+1)+b-a-(k+1))/(b-k))「(ab+b-a-k)/(2(n-1))」-(n-1)/(b-k)(「(an+b-a-k)/(2(n-1))」)2-1,那么G有一个[a,b]-因子F使得E(H)(∈)E(F).类似地,也得到了关于图G有一个r-因子含有G中给定的k-正则子图的度条件.进一步,指出这些度条件是最佳的.     

关于图的一种新分解

Alavi等人在[1]中定义了图的一种新分解,即升分解,并提出猜想:     

Greedily constructing Hamiltonian paths, Hamiltonian cycles and maximum linear forests

There are various greedy schemas to construct a maximal path in a given input graph. Associated with each such schema is the family of graphs where it always results a path of maximum length, or a Hamiltonian path/cycle, if there exists one. Considerable amount of work has been carried out, regarding the characterization and recognition problems of such graphs. We present here a systematic listing of previous results of this type and fill up most of the remaining empty entries.     

On the Ascending Subgraph Decompositions Problem

Inthispaper,weconsideronlysimpleundirectedgraphsandfollowBondyandMurtyl7]forterminologyandnotationnotdefinedhere.Y.Alaviandothershavegiventhedefinitionoftheascendingsubgraphdecomposionin[1].LetGbeagraphofqedgessatisfy( 1)5q<( ,).ThenGissaidtohavean...     

On the Ascending Subgraph Decomposition Problem

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Subgraph statistics in subcritical graph classes

Let H be a fixed graph and a subcritical graph class. In this paper we show that the number of occurrences of H (as a subgraph) in a graph in of order n, chosen uniformly at random, follows a normal limiting distribution with linear expectation and variance. The main ingredient in our proof is the analytic framework developed by Drmota, Gittenberger and Morgenbesser to deal with infinite systems of functional equations [Drmota, Gittenberger, and Morgenbesser, Submitted]. As a case study, we obtain explicit expressions for the number of triangles and cycles of length 4 in the family of series‐parallel graphs. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 631–673, 2017     

k-Factors and Spanning Subgraph in Graphs

In this paper, we discussed k-factors and spanning subgraph, and propose a conjecture which will lead to a series of important conclusion.     

关于图的升分解的Alavi猜想

Y.Alavi等人在1987年定义了图的一种新分解,即“升分解”(ascebding subgraph decomposition),并提出猜想:设自然数n≥2,G是由k个分离的星S_1,S_2,…,S_k构成的图,S_i含有a_i条边,n≤a_i≤2n-2,,则G可升分解为星的并。本文证明了当n=2k+i(i=0,1,2)时猜想成立。     

Heuristics for the Maximum Outerplanar Subgraph Problem

Determining the maximum outerplanar subgraph of a given graph is known to be an NP-complete problem. In the literature there are no earlier experiment on approximating the maximum outerplanar subgraph problem. In this paper we compare solution quality and running times of different heuristics for finding maximum outerplanar subgraphs. We compare a greedy heuristic against a triangular cactus heuristic and its greedy variation. We also use the solutions from the greedy heuristics as initial solutions for a simulated annealing algorithm.The main experimental result is that simulated annealing with initial solution taken from the greedy triangular cactus heuristic yields the best known approximations for the maximum outerplanar subgraph problem.Work funded by the Tampere Graduate School in Information Science and Engineering (TISE) and supported by the Academy of Finland (Project 51528).     

Dense Packing of Patterns in a Permutation

We study the length L _k of the shortest permutation containing all patterns of length k. We establish the bounds e ⁻² k ² < L _k ≤ (2/3 + o(1))k ². We also prove that as k → ∞, there are permutations of length (1/4 + o(1))k ² containing almost all patterns of length k. Received January 2, 2007     

The Dual Group of a Dense Subgroup

Throughout this abstract, G is a topological Abelian group and $\hat G$ is the space of continuous homomorphisms from G into the circle group ${\mathbb{T}}$ in the compact-open topology. A dense subgroup D of G is said to determine G if the (necessarily continuous) surjective isomorphism $\hat G \to \hat D$ given by $h \mapsto h\left| D \right.$ is a homeomorphism, and G is determined if each dense subgroup of G determines G. The principal result in this area, obtained independently by L. Außenhofer and M. J. Chasco, is the following: Every metrizable group is determined. The authors offer several related results, including these. 1. There are (many) nonmetrizable, noncompact, determined groups. 2. If the dense subgroup D _i determines G _i with G _i compact, then $ \oplus _i D_i $ determines Π_i G _i. In particular, if each G _i is compact then $ \oplus _i G_i $ determines Π_i G _i. 3. Let G be a locally bounded group and let G ⁺ denote G with its Bohr topology. Then G is determined if and only if G ⁺ is determined. 4. Let non $\left( {\mathcal{N}} \right)$ be the least cardinal κ such that some $X \subseteq {\mathbb{T}}$ of cardinality κ has positive outer measure. No compact G with $w\left( G \right) \geqslant non\left( {\mathcal{N}} \right)$ is determined; thus if $\left( {\mathcal{N}} \right) = {\mathfrak{N}}_1 $ (in particular if CH holds), an infinite compact group G is determined if and only if w(G) = ω. Question. Is there in ZFC a cardinal κ such that a compact group G is determined if and only if w(G) < κ? Is $\kappa = non\left( {\mathcal{N}} \right)?\kappa = {\mathfrak{N}}_1 ?$      

图模式挖掘中的子图同构算法

图模式挖掘问题在Web挖掘、生物信息学、社会关系等众多领域有广泛的应用,它涉及到子图的搜索以及子图的同构问题.这两个问题都具有相当高的计算复杂度,现有的子图同构问题大多采用最小编码算法,但对无标签图特别是对无标签无向图,该算法效率较底,从而子图的同构成为图模式挖掘问题的一个瓶颈.针对无标签图,以代数理论为基础,分别利用度序列和特征值构造了两种子图同构算法,用于对有向图和无向图的同构判别.最后对2个真实生物网络进行了仿真实验,结果表明,算法的效率优于现有算法.     

First-Order Complexity of Subgraph Isomorphism via Kneser Graphs

Mathematical Notes - The problem of finding the least number of variables of a first-order formula expressing the statement that an input-graph contains a subgraph isomorphic to a given...