浅谈算法实例在图论教学中的作用

摘要! 指出许多算法问题都直接或间接地与图有关 列举了一些常见的图论算法实例%主要包括经典算法实 例&可化为图论问题的算法实例以及变换图等%探讨图论教学中的一些常见算法实例问题福建省自然科学基金资助项目YZ.-!..!!

集合的基数与一一对应原理

集合的基数是学习实变函数论这一课程的重点和难点。深入地理解一一对应原理是学好这一内容的关键所在

给定最大匹配数的树的零阶广义Randi指标

图G的零阶广义Randi指标定义为0Rα(G)=v∈V(G)d(v)α,其中d(v)为图G的顶点v的度,α为任意实数.研究了树的零阶广义Rα指标的极值问题,利用分析和图的理论,确定了任意给定最大匹配数的树的最大和最小Rα的值,并刻画了达到该极值的树

具有最大Wiener-Hosoya指标的树

为进一步研究分子的性质,M.Randic在文[2]中首先介绍了Wiener-Hosoya指标.Wiener-Hosoya指标能够用h(T)=∑e∈E(T)(h(e)+h[e])组合原理来表示,我们是去计算Wiener指标W而不是去计算Hosoya topologic指标Z.本文主要讨论对于给定的顶点数时具有最大的Wiener-Hosoya指标的树

三角格上的σ全一问题

主要研究三角格上的σ全一问题.应用图论知识,利用数学归纳法,分别给出了以三角形,菱形(四边形)和正六边形为边界的三角格上的σ全一问题无解的充要条件,并在证明中给出有解情况下详细解的刻画

Enumerating tree-like polyphenyl isomers

NSFC [10831001]Enumeration of molecules is one of the fundamental problems in bioinformatics and plays an important role in drug discovery, experimental structure elucidation (e.g., by using NMR or mass spectrometry), molecular design and virtual library construction. We consider the enumeration of tree-like polyphenyls (C(6)nH(4n+2)). For this purpose, we de fine two generating functions T (x) and R (x) involving the numbers t(n) and r(n) of tree-like polyphenyls (TL-polyphenyls) and monosubstituted tree-like polyphenyls (MTL-polyphenyls), respectively. By characterizing the symmetry groups with respect to TL-polyphenyls and MTL-polyphenyls, we establish two functional equations for these two generating functions. This yields for the first time an efficient recursion formula for calculating the numbers t(n) and r(n). The two functional equations are also the fundamentals for analyzing their asymptotic behaviors, from which we derive the precise asymptotic values for both r(n) and t(n). The resulting asymptotic values are shown to fit well to the numerical results obtained by using our recursion formula. Finally, we give an explicit enumerating expression for TL-polyphenyls of a particular type: the linear polyphenyls

Dimer coverings on random multiple chains of planar honeycomb lattices

We study dimer coverings on random multiple chains. A multiple chain is a planar honeycomb lattice constructed by successively fusing copies of a 'straight' condensed hexagonal chain at the bottom of the previous one in two possible ways. A random multiple chain is then generated by admitting the Bernoulli distribution on the two types of fusing, which describes a zeroth-order Markov process. We determine the expectation of the number of the pure dimer coverings (perfect matchings) over the ensemble of random multiple chains by the transfer matrix approach. Our result shows that, with only two exceptions, the average of the logarithm of this expectation (i.e., the annealed entropy per dimer) is asymptotically nonzero when the fusing process goes to infinity and the length of the hexagonal chain is fixed, though it is zero when the fusing process and the length of the hexagonal chain go to infinity simultaneously. Some numerical results are provided to support our conclusion, from which we can see that the asymptotic behavior fits well to the theoretical results. We also apply the transfer matrix approach to the quenched entropy and reveal that the quenched entropy of random multiple chains has a close connection with the well-known Lyapunov exponent of random matrices. Using the theory of Lyapunov exponents we show that, for some random multiple chains, the quenched entropy per dimer is strictly smaller than the annealed one when the fusing process goes to infinity. Finally, we determine the expectation of the free energy per dimer over the ensemble of the random multiple chains in which the three types of dimers in different orientations are distinguished, and specify a series of non-random multiple chains whose free energy per dimer is asymptotically equal to this expectation.NSFC [10831001, 11061027

纳滤法分离回收咪唑离子液体研究

离子液体作为一种结构可调的绿色试剂已经在许多领域被广泛研究和应用。针对离子液体大规模工业化应用，需要解决的最突出问题之一就是如何高效地回收重复利用，因此，开发离子液体回收方法具有重要现实意义。纳滤作为一种重要的化工分离单元操作，由于其操作简单，低温节能等优点，被广泛应用于多种体系中各组分的分离和提纯。本论文开展了纳滤分离一系列咪唑离子液体的研究。考察了分子大小以及溶液环境对膜分离效果的影响。重点分析了离子液体团簇对膜分离效果，以及膜污染和膜重复使用的影响。同时，基于团簇对溶解扩散模型进行了合理改进，以适用于不同链长的咪唑离子液体水溶液。此外，考察了纳滤耦合蒸馏提纯离子液体的理论经济性，提出了更加高效节能地回收离子液体方法。具体内容如下：（1）使用陶氏纳滤膜分离了一系列1-烷基-3-甲基咪唑盐（[Cnmim]X，n表示阳离子烷基侧链数，X表示阴离子）。考察了分子大小对膜通量（Jv）和截留率（Ro）的影响。同时研究了溶液环境对膜分离效果的影响，包括溶液pH，离子强度，大分子有机污染物。实验结果发现，分子大小和大分子有机污染物通过影响孔径筛分效应影响膜分离效果，pH和离子强度通过影响电荷效应影响膜分离效果。此外，溶液环境也对某些离子液体的物化性质产生影响，从而进一步影响膜分离机制及膜分离效果。（2）研究了离子液体团簇对膜分离效果的影响。使用陶氏纳滤膜分离了一系列不同浓度的长碳链离子液体水溶液。通过实验和分子动力学模拟方法，研究了团簇在水溶液中的存在状态。阐述了团簇对膜通量和截留率的影响，以及对膜污染和膜重复利用的影响。实验结果发现，团簇主要通过影响孔径筛分作用影响膜分离效果：离子液体浓度增加，团簇程度提高，团簇的平均大小越大，导致膜通量降低，截留率升高。此外，团簇缓解膜污染程度，其原因是团簇进入孔道的难度更大，空间位阻效应得到了加强，膜孔道被堵塞几率下降。此外，通过调控团簇大小，提高了纳滤分离长碳链离子液体水溶液的分离性能。（3）考察了溶解扩散模型对纳滤分离离子液体水溶液的适用性。使用陶氏纳滤膜分离了不同浓度的长碳链离子液体水溶液，测定了它们的膜通量和截留率。使用溶解扩散模型对截留率进行了预测，并和实验值进行了对比。结果发现，该模型适用于低浓度长碳链离子液体水溶液，而不适用于高浓度长碳链离子液体水溶液。建立了考虑团簇因素的溶解扩散模型，通过分子动力学模拟，对不同浓度的离子液体水溶液中的团簇大小进行了合理地定量计算，并依据团簇大小对该模型中的相关参数进行了合理地修饰。计算结果发现，相比未改进的模型，通过改进后的模型预测地计算值和实验值更加吻合。（4）分析了纳滤耦合蒸馏提纯离子液体的理论经济性。计算了纳滤联合减压蒸馏方法（先经纳滤预浓度，再减压蒸馏）提纯离子液体的成本，并和单纯减压蒸馏方法提纯离子液体的成本进行了对比。实验结果发现，纳滤联合减压蒸馏方法的成本大大低于单一减压蒸馏的成本，其原因是气体做功能耗远远低于加热蒸发的能耗。同时，基于团簇对膜分离性能的影响，提出了更加高效节能地回收长碳链离子液体的方法，即二级纳滤耦合减压蒸馏提纯长碳链离子液体的方法。首先使用孔径较小的纳滤膜进行脱水，随着离子液体浓度升高，团簇生成并逐渐变大，适时改用孔径较大的膜纳滤继续进行脱水，最后经过两步预浓缩的离子液体经减压蒸馏彻底除水。实验结果发现，该方法相比一级纳滤，成本得到进一步降低。;Ionic liquis (ILs) have been extensively studied in many fields as a structurally tunable green reagent. In terms of large-scale application of ILs, one of the most prominent obstacles is how to efficiently recycle ILs. Therefore, it has important practical significance to develop methods to recover ILs. As an important chemical separation unit, nanofiltration is widely used in the separation and purification fields due to its simple operation and energy saving. In this thesis, a series of imidazole-based ILs were filtrated by nanomembranes, and the effects of molecular size and solution environment on membrane filtration have been studied. What&rsquo;s more, the effect of ion cluster on membrane filtration, membrane fouling and membrane re-use were also investigated. the solution diffusion model was rationally improved based on ion cluster, which makes it suitable for aqueous solutions of ILs with different chain lengths. In addition, the theoretical economic analysis of nanofiltration combined distillation of ILs was studied, and a more efficient and energy-saving method for recovering long-alkyl-chain ILs was proposed. The details are as follows:Firstly, A series of 1-alkyl-3-methylimidazolium-based ILs ([Cnmim]X, n represents the number of cationic alkyl side chains, and X represents anion) were filtrated using Dow nanofiltration membranes. The effect of molecular sizes on volumetric membrane flux (Jv) and observed rejection (Ro) was studied, such as the size of the cationic alkyl side chain and the different halide anions. In addition, the effect of solution environment on membrane filtration was investigated, including pH, ionic strength, and macromolecular organic pollutants. The experimental results showed that the molecular size and macromolecular organic pollutants influence the nanofiltration performance by impacting sieving effect, and the pH and ionic strength influence the nanofiltration performance by affecting charge effect. Besides, the solution environment also impacts the physicochemical properties of certain ILs, thereby influencing the nanofiltration mechanism, and thus the nanofiltration performance.Secondly, the effect of ion cluster on concentration of ILs aqueous solution by nanofiltration was investigated. A series of long-alkyl-chain ILs aqueous solutions with different concentrations were filtrated using Dow nanofiltration membranes. The existential state of ion cluster in aqueous solution was studied by experiments and molecular dynamics simulation. The effect of ion cluster on Jv, Ro, membrane fouling and membrane recycling were investigated. The experimental results showed that ion cluster mainly influences the nanofiltration performance by affecting sieving effect: with the increasing of concentration of ILs, the degree of ion cluster increases, and thus the larger average size of ion cluster, which leads to the decrease of the Jv and the increase of the Ro. In addition, ion cluster alleviates the degree of membrane fouling, It is more difficult for ion cluster to enter membrane pores, because the steric hindrance effect is enhanced, and the probability of the membrane pores being blocked is decreased. What&rsquo;s more, according to the influence of the cluster on the membrane filtration, the filtration performance was enhanced by adjust the molecular size of ion cluster in long-alkyl-chain ILs aqueous solutions.Third, The applicability of the solution-diffusion model to the nanofiltration of ILs aqueous solutions was investigated. A series of long-alkyl-chain ILs aqueous solutions with different concentrations were filtrated using Dow nanofiltration membranes, and their Jv and Ro were measured. Using the solution-diffusion model, Ro was predicted by the experimental value Jv. As a result, it was found that the model is suitable for long-alkyl-chain ILs aqueous solutions with low concentrations, but not for long-alkyl-chain ILs aqueous solutions with high concentrations. The solution-diffusion model was reasonably modified based on ion cluster. The molecular size of ion clusters in different concentrations of ILs was quantitatively calculated by molecular dynamics simulation. The relevant parameters in the model were calculated according to the size of ion cluster. The calculation results showed that compared with the unmodified model, the calculated values predicted by the improved model agreed better with the experimental values.Fourth, The theoretically economic investigation of nanofiltration combined with distillation to recover ILs was investigated. The costing of nanofiltration combined with vacuum distillation (pre-concentration by nanofiltration and followed by reduced pressure distillation) for recovering ILs aqueous solutions was calculated. The experimental results showed that the costing of the nanofiltration combined with vacuum distillation method is much lower than that of simple vacuum distillation. The reason for the reduction in costing is that the gas work is much lower than the energy consumption of heating and evaporation. In addition, based on the influence of ion cluster on membrane filtration, a more efficient and energy-saving method for recovery of long-alkyl-chain ILs was proposed, which is a method of two-stage nanofiltration combined with vacuum distillation to purify long-alkyl-chain ILs. Firstly, the membrane with smaller pore size is used for filtration. As the concentration of ILs increases, the ion clusters are formed and the size gradually becomes larger, and the membrane with a larger pore diameter is used at the appropriate moment. Finally, the pre-concentrated ILs was completely purified by reduced pressure distillation. The experimental results show that the costing of the method is further reduced compared to the one-stage nanofiltration.&nbsp;</p

The Diameter of Interchange Graphs and the Brualdi's Conjecture

设Ｒ＝（ｒ１，ｒ２…ｒｍ）及 Ｓ＝（Ｓ１，Ｓ２，…，Ｓｎ）为两个正整数向量，满足Σｍｉ＝１ｒｉ＝Σｎｊ＝１ｓｊ＝ Ｋ．记Ｇ（Ｒ，Ｓ）为（０，１）－矩阵类 Ｕ（Ｒ，Ｓ）的变换图．Ｂｒｕａｌｄｉ在文山中给出了 Ｇ（Ｒ，Ｓ）的直径厂（Ｇ（Ｒ，Ｓ））的一个上界：ｍｎ／２－１，并猜想Ｄ（Ｇ（Ｒ，Ｓ））≤ｍｎ／４．本文通过对有向图围长的研究得到了Ｄ（Ｇ（Ｒ，Ｓ））的一个新的上界：１／２ｍｎ－１／６ｔ（ｔ－１）（４ｔ＋１），其中Ｔ＝　　．Let R = (r1, r2，…，rm) and S = (s1, s2,..., sn) be two nonnegative integral vectors with Σmi=1ri=Σnj=1sj=K. Denote by G(R. S) the interchange graph of the class U(R, S) of (0,1)-matrices. Brualdi proved that the diameter of G(R, S) is less than mn/2 - 1, and further conjectured that it is no more than mn/4. In this paper, by studying the girth of the directed graphs, a new upper bound of the diameter of G(R, S) is proved to be 1/2mn - 1/6t(t - 1)(4t + 1), where t = mn/(m + n).厦门大学博士自选课题资助项目（Y07001

The Edge Connect ivity of Generalized In terchange Graph G+ ∞ (R , S )

摘要　记¡+ ∞ (R , S ) 为具有行和向量R 及列和向量S 的所有m ×n 阶非负整数矩阵的集 合. 广义变换图G+ ∞ (R , S ) 的顶点定义为¡+ ∞ (R , S ) 中的矩阵, 两个顶点(矩阵) 相邻当且仅当它们 可通过一次变换相互得到. 并证明G+ ∞ (R , S ) 的边连通度等于其顶点的最小度D(G+ ∞ (R , S ) ). Abs tra c t　L et ¡+ ∞ (R , S ) deno te the class of allm ×n in tegralm at rices having row sum vecto r R and co lum n sum vecto r S. The generalized in terchange graph G+ ∞ (R , S ) is the graph w here the vert ices are them at rices in ¡+ ∞ (R , S ) and two vert ices are adjacen t p rovid2 ed they differ by an in terchange. It is show n that the edge connect ivity of G+ ∞ (R , S ) is equal to it s m in im um degree of vert ices