On Locating Numbers of Graphs

Let G =( V,E) be a connected graph and W = { w_1,w_2,…,w_k} be an ordered subset of V( G).For any vertex v ∈V,the locating code of v with respect to W is the k-vector CW( v) = { d( v,w_1),d( v,w_2),…,d( v,w_k) },W is said to be a locating set of G if distinct vertices have the distinct locating code,and the locating number of G is defined as: Loc( G) = min{ | W| : W is a locating set of G}.We study the locating set and locating number of a graph G,obtain some bounds for the locating numbers of graphs,and determine the exact value of Loc( G) for some special classes of graphs,such as cycles,wheels,complete t-partite graph and some Cartesian products of paths and cycles. In addition,we also prove that Loc( T) ≥Δ-1 holds for all trees T with maximum degree Δ,and shows a tree T with Loc( T) = Δ-1.     

Optimal Orientations of the Generalized Cycles

Optimal orientations of the generalized cycles are studied. For a graph G, let D（G） be the family of the strong orientations of G,d（G）=min {d（D） D∈D（G） and ρ（G）=d（G）-d（G）, whered（G） and d （D） are the diameters of G and D respectively. Evaluate the value of ρ（G） is evaluated by reduction to absurdity when G is a generalized cycle Cn [Km], and a complete result is obtained.     

A two-stage flexible flow-shop scheduling problem with m identical parallel machines on one stage and a batch processor on the other stage

This paper considers a hybrid two-stage flow-shop scheduling problem with m identical parallel machines on one stage and a batch processor on the other stage. The processing time of job Jj on any of m identical parallel machines is aj≡a （j∈N）, and the processing time of job Jj is bj（j∈N） on a batch processorM. We take makespan （Cmax） as our minimization objective. In this paper, for the problem of FSMP-BI （m identical parallel machines on the first stage and a batch processor on the second stage）, based on the algorithm given by Sung and Choung for the problem of 1 ｜ri, BI｜Cmax under the constraint of the given processing sequence, we develop an optimal dynamic programming Algorithm H1 for it in max {O（nlogn）, O（nB）} time. A max {O（nlogn） , O（nB）}time symmetric Algorithm H2 is given then for the problem of BI-FSMP （a batch processor on the first stage and m identical parallel machines on the second stage）.     

Bondage and Reinforcement Number of γf for Complete Multipartite Graph

The bondage number of γf, bf(G) , is defined to be the minimum cardinality of a set of edges whose removal from G results in a graph G′ satisfying γf(G′)＞γf(G). The reinforcement number of γf, rf(G), is defined to be the minimum cardinality of a set of edges which when added to G results in a graph G′ satisfying γf(G′)＜γf(G). G.S.Domke and R.C.Laskar initiated the study of them and gave exact values of bf(G) and rf(G) for some classes of graphs. Exact values of bf(G) and rf(G) for complete multipartite graphs are given and some results are extended.     

Removal of anionic ions from single material solution by bauxite tailings modified with FeCl3·6H2O

The adsorbabilities of the unmodified and modified bauxite tailings for Cr（Ⅵ）, As（Ⅴ） and F（Ⅰ） ions were investigated. Batch experiments were carried out to determine the removal rate as a function of adsorbent dosage, solution pH value and shaking time. The results show that the maximum removal rates of Cr（Ⅵ）, As（Ⅴ） and F（Ⅰ ） are respectively 99%, 99% and 90% by using the modified bauxite tailings. The isoelectric point of the unmodified bauxite tailings is 3.6, and that of the modified bauxite tailings is 5.0, which shifts to lower pH values in Cr（Ⅵ） solution. This indicates a specific adsorption of the anionic species on the modified bauxite railings. A new band of Cr2O72 appears in the FTIR, showing that Cr（Ⅵ） is adsorbed on the modified bauxite railings in the form of chemistry adsorption. The adsorption data of Cr（Ⅵ） on the modified bauxite tailings are well described by Freundlich model. The investigations of kinetic models show that pseudo-second-order kinetic model provides the best correlation for the experimental data.     

On Minus Paired－Domination in Graphs

The study of minus paired-domination of a graph G = ( V, E) is initiated. Let S lontain in V be any paired-dominating set of G, a minus paired-dominating function is a function of the form f: V→ { - 1, 0, ｝such that f(υ) = 1 for υ∈S, f(υ)≤0 for υ∈V- S, and f(N[υ])≥l for all υ∈V. The weight of a minus paired-dominating function f is ω(f)=∑f(υ), over all vertices υ∈V. The minus paired-domination number of a graph G is γp^-(G)= min｛ω (f)| f is a minus paired-dominating function of G｝. On the basis of the minus paired-domination number of a graph G defined, some of its properties are discussed.     

邻域并与最小度给出的哈米顿图的充分条件

0 IntroductionA cycle (path) in a graph G is called a hamiltonian cycle (path) in G if it con-tains all the vertices of G.A graph is hamiltonian (traceable) if it has a hamiltoniancycle (path).The neighborhood of a vertex v,denoted N(v),is the set of all verticesadjacent to v.We define the distance between two vertices u and v,denoted dist(u,v),as the minmum of the lengths of all u-v paths.Let NC2=min|N(u)∪N(v)|,DC2=min{max{d(u),d(v)}},where the minimum is taken over all pairs of vertices u,vthat are at distance two in the graph.Let δrepresent the minimum degree of G.Referto [5] for other terminology.     

Effect of Typical Elements on the Heredity of Hot-rolled and Annealed Texture in Non-oriented Electrical Steel

Macroscopic texture and microscopic orientation in hot-rolled and annealed sheets of nonoriented electrical steel were studied by XRD and EBSD techniques. The microstructure of hot-rolled and annealed samples was studied by OM. Experimental results indicate that a strong heredity is observed in texture evolution between hot-rolled texture and annealed texture. Typical elements have a large effect on the recrystallization microstructure and texture distribution. The texture distribution through thickness is highly affected by recrystallization in hot rolled sheets. The recrystallization is boosted by Si and Al. Goss grains originate from cracked initial 100 columnar grains. {110}112, {112}111 and {111}112 grains are related to Goss grains. In subsurface lay of hot rolled sheets, Al can strengthen Goss texture and weaken copper-type texture. {112}111 texture and {110}112 texture are strengthened by Si. In the central layers, {100}001 texture and {111}121 texture are weakened by Al. {100}011 texture is increased by Al. Si can increase the proportion of γ-fiber texture and decrease that of {100}011 texture. In annealed texture, {100}001 texture and Goss texture are decreased by Al and Si. γ-fiber texture is increased by Si and {111}121 texture is preferentially increased by Al. The recrystallized grain size is increased and iron loss of annealed sheets is reduced by Al and Si, which means that the magnetic properties are optimized by the Al and Si content.     

Julia sets of semi-conjugated transcendental entire functions

Suppose that f and g are two transcendental entire functions, and h is a non-constant periodic entire function. We denote the Julia set and Fatou set off by J（f） and F（f）, respectively, lffand g are semiconjugated, that is, h · f = g · h, in this paper, we will show that z ∈ J（f） if and only if h（z） ∈ J（g） （ similarly, z F（f） if and only ifh（z） ∈ F（g））, and this extends a result of Bergweiler.     

Microstructure and formation mechanism of twins of laths of austenite with high nitrogen

The microstructure of composite diffusion layer of the nitrided and chromized 0.2% carbon steel is investigated using TEM and EDS. It is found that laths of austenite with high nitrogen (λN) precipitate from α-ferrite matrix in the deeper zone of the diffusion layer. These λN laths are all twins, with their {111} twinning planes parallel to the lath axis, thus forming a characteristic "back-to-back" morphology. There are two types of λN lath. The first is a genuinely {111} twin, and λN and α keep the accurate K-S relationship, and each λN and α form a sharp and smooth λN/α interface of {335}λN//{341}a, namely habit plane {335}fcc. The second is a pseudo-twin, with micro-twins {111} or faults formed within the two twin components. Localized lattice deformation (relaxation) seems to have occurred at the interfaces of the second type of λN due to the formation of micro-twins or faults within the twin components. These micro-twins or faults make the orientation relationship (OR) between each of the λN and the (-matrix deviate from the accurate K-S OR, and the OR between two λN twin components deviate from the genuine {111} twin relationship. In addition, the λN/α interface of the second type of λN is not as sharp or smooth as that of the first one.     

边色临界图的1-因子和几乎1-因子的存在性

根据Vizing邻接引理和关于临界图和二分图的3个结论,利用图的1-因子和几乎1-因子存在的充要条件,采用结构图论的方法证明了:1)若G是2n阶临界图,且δ(G)≥n-3,则G存在1-因子;2)若G是2n+1阶临界图,且δ(G)≥n-4,则G存在几乎1-因子.     

关于色唯一图的注记

设P(G;λ)是图G的色多项式,如果对任意图H,当P(H;λ)=P(G;λ)时,都有H和G同构,则称图G是色唯一的。本文给出了由两个块H和K2构成的图G是色唯一的当且仅当H是色唯一点可迁的。     

关于正则图的路分解

在3-正则图的{P3,P4}分解的基础上,结合偶次图的圈分解,证明了任意的4-正则图存在{P4,P5}分解,任意的5-正则图存在{P5,P6}-的分解。同时还提出了k-正则图路分解的猜想。     

关于图的两类强符号控制数的下界

设图G=(V,E)为无孤立点的简单图,且f:V→{-1,1}为G上的一个函数,如果对于任意的顶点v∈V,均有f[v]≥2,则称f是图G的一个强符号控制函数。图G的强符号控制数定义为γss(G)=min{w(f)|f是图G的强符号控制函数}。设k是1≤k≤|V|的正整数,f:V→{-1,1}为图G上的一个函数,如果在图G中至少有k个顶点,使得f[v]≥2,则称f是图G的一个强k-符号控制函数。图G的强k-符号控制数定义为γkss=min{w(f)|f是图强G的k-符号控制函数}。分别得出了强符号控制数及强k-符号控制数的几种形式的下界。     

梯子的模和数

1990年,F．Harary提出了和图的概念,模和图和模和数的概念是由Boland、Sutton等人提出来的．模和图是取S(?)Zm＼{0}且所有算术运算均取模m(≥|S|+1)的和图,其中Zm={0,1,2,…,m-1}．一个图G的模和数ρ(G)是使得G∪ρK1是模和图的孤立点数ρ的最小值．本文证明了kL3(k≥2)是模和图,因而也是模整和图．     

Sm∨Fn的全色数

设G（V,E）是阶数至少是2的简单连通图,k是正整数,若厂是从V（G）∪E（G）到{1,2,…,k}的一个映射,使得：对于任意的uv,vw∈E（G）,u≠w,有f（uv）≠f（vw）;且对于任意的uv∈E（G）,u≠v,有f（u）≠f（v）,f（u）≠f（uv）,f（v）≠f（uv）,则称f为G的一个k-全染色（简记成k-TC of G）．而Xt（G）=min{k｜k—TC of G},称为G的全色数．设G和H是点边都不相交的简单图,V（G∨H）=V（G）∪V（H）,E（G∨H）=E（G）∪E（H）∪{uv｜u∈V（G）,v∈V（H）},则称G∨H是G与H的联图。给出m＋1阶星和n＋1阶扇的联图的全色数。     

点可区别全色数的一个上界

设G是简单图,f是从V（G）UE（G）到{1,2,…,k）的一个映射．对每个u∈y（G）,令c（u）={f（u）}v∈V（G）,uv∈ E（G）}．如果,是k-正常全染色,且对任意u,v∈V（G）（u≠v）,有c（u）≠c（v）,那么称f为图G的k-点可区别全染色（简记为k-VDTC）．数χvt（G）=min{k｜G-有k—VDTC}称为图G的点可区别全色数．通过应用概率方法,证明了对任意最大度A≥2的图G,χvt（G）≤32（△＋1）．     

图的多数控制数的下界

设G=(V,E)是简单图,V表示G的顶点集,E表示G的边集.对任何实值函数f∶V→R和V的子集S,令f(S)=∑u∈Sf(u).设f∶V→{-1,1}是G上的一个函数.如果对于V的至少一半的顶点v,f(N[v])≥1,则称f是G上的多数控制函数.图G的多数控制数是γmaj(G)=min{f(V)|f是G上的一个多数控制函数}.得到了这个参数的下界,推广了Henning的一些结果.     

二维离散型随机变量相互独立

二维离散型随机变量（Ｘ，Ｙ）相互独立的定义是：Ｆ（ｘ，ｙ）＝ＦＸ（ｘ）ＦＹ（ｙ）。其中Ｆ（ｘ，ｙ），ＦＸ（ｘ），ＦＹ（ｙ）分别为（Ｘ，Ｙ），Ｘ，Ｙ的分布函数。一般用等式Ｐ｛Ｘ＝ｘｉ，Ｙ＝ｙｊ｝＝Ｐ｛Ｘ＝ｘｉ｝Ｐ｛Ｙ＝ｙｊ｝进行判定，其中（ｘｉ，ｙｊ）为（Ｘ，Ｙ）所有可能取值，ｉ＝１，２，…；ｊ＝１，２，…；没有给出具体证明。本文给出二维离散型随机变量相互独立的定义及与这种判定方法等价的严格证明。