Nanosized Carbon Macrocycles Based on a Planar Chiral Pseudo Meta-[2.2]Paracyclophane

Structural designs combining cycloparaphenylenes (CPPs) backbone with planar chiral [2.2]paracyclophane ([2.2]PCP) lead to optical-active chiral macrocycles with intriguing properties. X-ray crystal analysis revealed aesthetic necklace-shaped structures and size-dependent packages with long-range channels. The macrocycles exhibit unique photophysical properties with high fluorescence quantum yield of up to 82 %, and the fluorescent color varies with ring size. In addition, size-dependent chiroptical properties with moderately large CPL dissymmetry factor of 10⁻³ and CPL brightness in the range of 30–40 M⁻¹ cm⁻¹ were observed.     

Existential monadic second order logic of undirected graphs: The Le Bars conjecture is false

In 2001, J.-M. Le Bars disproved the zero-one law (that says that every sentence from a certain logic is either true asymptotically almost surely (a.a.s.), or false a.a.s.) for existential monadic second order sentences (EMSO) on undirected graphs. He proved that there exists an EMSO sentence ? such that $\mathsf{P}(G_n??)$ does not converge as $n\to \infty$ (here, the probability distribution is uniform over the set of all graphs on the labeled set of vertices $\{1,\dots ,n\}$). In the same paper, he conjectured that, for EMSO sentences with 2 first order variables, the zero-one law holds. In this paper, we disprove this conjecture.     

Control of Axial Chirality by Planar Chirality Based on Optically Active [2.2]Paracyclophane

Optically active X-shaped molecules based on the planar chiral [2.2]paracyclophane building block were prepared, in which di(methoxy)terphenyl units were stacked on the central benzene rings. At 25 °C, anisolyl rings freely rotate in solution, while in the crystal form, they are fixed by intramolecular CH–π interactions, thereby leading to the expression of the axial chirality, i.e., propeller chirality was exhibited by the planar chiral [2.2]paracyclophane moiety. The X-shaped molecule exhibited good circularly polarized luminescence (CPL) profiles with moderate Φ_PL and a large g_lum value in the order of 10⁻³ at 25 °C, in solution. In contrast, at −120 °C, dual CPL emission with opposite signs was observed. According to the theoretical studies, the rotary motion of the anisolyl units is suppressed in the excited states, and so emission from two isomers could be observed. These results demonstrate that the axial chirality was controlled by the planar chirality, leading ultimately to propeller chirality.     

No optimal 1‐planar graph triangulates any nonorientable closed surface

Suzuki [Discrete Math. 310 (2010), 6–11] proved that for any orientable closed surface F² other than the sphere, there exists an optimal 1‐planar graph which can be embedded on F² as a triangulation. However, for nonorientable closed surfaces, the existence of such graphs is unknown. In this article, we prove that no optimal 1‐planar graph triangulates a nonorientable closed surface.     

基于不同条件的旅游路线规划问题研究

运用2015年全国研究生数学建模竞赛F题的数据资料,针对旅游路线合理规划问题的第一问展开研究.以F题的问题一为起点进行了分析研究,是因为第一问的完成是解决后续问题的关键.首先通过地图搜集并补全了缺失数据并对数据进行合理的处理,然后采用将旅游年数最少目标转化为该最小生成树的"最少圈覆盖"方法来进行对问题一的求解,也可以理解为一个广义的多旅行商问题,以旅行商的人数(即年数)最少为目标.采用Dijkstra算法、最少圈覆盖法、智能算法和图论聚类等方法,通过这些方法建立了单目标优化模型,并运用旅行商问题和模型之间的转换来对问题进行分析与求解.     

On weighted spectral radius of unraveled balls and normalized Laplacian eigenvalues

For a graph G, the unraveled ball of radius r centered at a vertex v is the ball of radius r centered at v in the universal cover of G. We obtain a lower bound on the weighted spectral radius of unraveled balls of fixed radius in a graph with positive weights on edges, which is used to present an upper bound on the $s^{\text{th}}$ (where $s\ge 2$) smallest normalized Laplacian eigenvalue of irregular graphs under minor assumptions. Moreover, when $s=2$, the result may be regarded as an Alon–Boppana type bound for a class of irregular graphs.     

There is no upper bound for the diameter of the commuting graph of a finite group

We construct a family of finite 2-groups which have commuting graph of increasing diameter.