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Given a graph sequence denote by T3(Gn) the number of monochromatic triangles in a uniformly random coloring of the vertices of Gn with colors. In this paper we prove a central limit theorem (CLT) for T3(Gn) with explicit error rates, using a quantitative version of the martingale CLT. We then relate this error term to the well-known fourth-moment phenomenon, which, interestingly, holds only when the number of colors satisfies . We also show that the convergence of the fourth moment is necessary to obtain a Gaussian limit for any , which, together with the above result, implies that the fourth-moment condition characterizes the limiting normal distribution of T3(Gn), whenever . Finally, to illustrate the promise of our approach, we include an alternative proof of the CLT for the number of monochromatic edges, which provides quantitative rates for the results obtained in [7]. 相似文献
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《Discrete Mathematics》2022,345(12):113069
The toughness of a noncomplete graph G is the maximum real number t such that the ratio of to the number of components of is at least t for every cutset S of G. Determining the toughness for a given graph is NP-hard. Chvátal's toughness conjecture, stating that there exists a constant such that every graph with toughness at least is hamiltonian, is still open for general graphs. A graph is called -free if it does not contain any induced subgraph isomorphic to , the disjoint union of and two isolated vertices. In this paper, we confirm Chvátal's toughness conjecture for -free graphs by showing that every 7-tough -free graph on at least three vertices is hamiltonian. 相似文献
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By 5-h reaction of cis-[RuIICl2(DMSO)4] (M2) with K10[α2-P2W17O61] (M3) in ice-cooled, HCl-acidic aqueous solution, a water-soluble 1:2-type diamagnetic ruthenium(II) complex of formula K18[RuII(DMSO)2(P2W17O61)2]·35H2O (M1) was unexpectedly obtained as an analytically pure, homogeneous tan-colored solid, in which two DMSO ligands are coordinated to the ruthenium(II) atom. The cytotoxic potential of the complex was tested on C33A, DLD-1, and HepG-2 cancer cells and human normal embryonic lung fibroblasts cell MRC-5; the viability of the treated cells was evaluated by MTT assay. The mode of cell death was assessed by morphological study of DNA damage and apoptosis assays. Compound M1 induced cell death in a dose-dependent manner, and the mode of cell death was essentially apoptosis though necrosis was also noticed. Cell cycle analysis by flow cytometry indicated that M1 caused cell cycle arrest and accumulated cells in S phase. 相似文献
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Let be a cycle of order 4. Write for the maximum number of edges in a balanced 3‐partite graph whose vertex set consists of three parts, each has vertices that have no subgraph isomorphic to . In this paper, we show that , where and is a prime number. Note that from Tait and Timmons's works. Since for every integer , one can find a prime such that , we obtain that . 相似文献
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受不可逆损失的影响,热光伏能量转换器件在高品位热能回收与利用方面受到限制.本文揭示不可逆损失来源,提供热光伏能量转换器件性能提升方案.利用半导体物理和普朗克热辐射理论,确定热光伏能量转换器件在理想条件下的最大效率.进而考虑Auger与Shockley-Reed-Hall非辐射复合和不可逆传热损失对光伏电池的电学、光学和热学特性的影响,预测热光伏器件优化性能.确定功率密度、效率和光子截止能量的优化区间.结果表明:相比于理想热光伏器件,非理想热光伏器件的开路电压、短路电流密度和效率有所降低;优化热光伏电池电压、光子截止能量和热源温度,可提升器件的功率密度和效率.通过对比发现理论与实验结果较一致,所得结果可为实际热光伏能量转换器件的研制提供理论指导. 相似文献
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ABSTRACTWe study the molecular-scale features of the solid surface that result in the spontaneous motion of a nanodroplet due to the periodic variation of temperature. We first employ a thermodynamic model to predict the variation of solid–fluid interfacial properties that can result in the above motion. The model identifies a composite (surface couple) made of two surfaces that are characterised by a large difference between the entropic parts of the solid–liquid interfacial free energies. In order to understand the molecular-scale features of the two surfaces that may form a surface couple, we performed grand canonical Monte Carlo simulations of Lennard Jones fluid and crystalline surfaces made of Lennard Jones-like atoms. We then used the cumulant expansions of the perturbation formulas to divide the interfacial entropy into two parts: The one that is directly affected by the solid–fluid attraction (direct part), and the other (indirect part) that is indirectly affected by the solid–fluid attraction via the alteration of interfacial fluctuations. Our results indicate that two surfaces form a surface couple if the differences between their chemical natures lead to large differences in the indirect part of the interfacial entropy, while the direct part remains relatively unaffected. 相似文献