1‐Factor and Cycle Covers of Cubic Graphs

Let G be a bridgeless cubic graph. Consider a list of k 1‐factors of G. Let be the set of edges contained in precisely i members of the k 1‐factors. Let be the smallest over all lists of k 1‐factors of G. Any list of three 1‐factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge‐covers and for the existence of three 1‐factors with empty intersection. Furthermore, if , then is an upper bound for the girth of G. We also prove some new upper bounds for the length of shortest cycle covers of bridgeless cubic graphs. Cubic graphs with have a 4‐cycle cover of length and a 5‐cycle double cover. These graphs also satisfy two conjectures of Zhang 18 . We also give a negative answer to a problem stated in 18 .     

Nonvertex‐Balanced Factors in Random Graphs

We prove part of a conjecture by Johansson, Kahn, and Vu (Factors in random graphs, Random Struct. Algorithms 33 (2008), 1, 1–28.) regarding threshold functions for the existence of an H‐factor in a random graph . We prove that the conjectured threshold function is correct for any graph H which is not covered by its densest subgraphs. We also demonstrate that the main result of Johansson, Kahn, and Vu (Factors in random graphs, Random Struct. Algorithms 33 (2008), 1, 1–28) generalizes to multigraphs, digraphs, and a multipartite model.     

On independent sets in random graphs

The independence number of a sparse random graph G(n,m) of average degree d = 2m/n is well‐known to be with high probability, with in the limit of large d. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size , i.e., about half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with size for any fixed (independent of both d and n). In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size k of the independent sets passes the point . Roughly speaking, we prove that independent sets of size form an intricately rugged landscape, in which local search algorithms seem to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independent sets. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 436–486, 2015     

关于图的同构判定方法的探讨

对于两图的同构的判定方法进行较深入的探讨,给出判定两图同构和判定两图不同构的几种方法,并对其判定方法的优劣进行比较.     

关于两个图的一类新连接图的谱

给定2个图G1和G2,设G1的边集E(G1)={e1,e2,?,em1},则图G1⊙G2可由一个G1,m1个G2通过在G1对应的每条边外加一个孤立点,新增加的点记为U={u1,u2,?,um1},将ui分别与第i个G2的所有点以及G1中的边ei的端点相连得到,其中i=?1,2,?,m1。得到：（i）当G1是正则图,G2是正则图或完全二部图时,确定了G1⊙G2的邻接谱（A-谱）。（ii）当G1是正则图,G2是任意图时,给出了G1⊙G2的拉普拉斯谱（L-谱）。（iii）当G1和G2都是正则图时,给出了G1⊙G2的无符号拉普拉斯谱（Q-谱）。作为以上结论的应用,构建了无限多对A-同谱图、L-同谱图和Q-同谱图;同时当G1是正则图时,确定了G1⊙G2支撑树的数量和Kirchhoff指数。     

Fundamental Groups of Graphs of Cyclic Subgroup Separable and Weakly Potent Groups

We give a characterization of the cyclic subgroup separability and weak potency of the fundamental group of a graph of polycyclic-by-finite groups and free-by-finite groups amalgamating edge subgroups of the form × D,where h has infinite order and D is finite.     

一种基于谱线幅值归零的加窗插值FFT谐波检测方法

提出了一种基于谱线幅值归零的加窗插值FFT谐波检测方法用于分析复杂电网信号中的各次谐波.该检测方法将每一次在离散频谱中找到的最大谱线及其左右两侧的数根谱线幅值归零,直到各次谐波分析完毕.通过仿真实验分析了在基波频率稳定时对模拟信号的测量精度,研究了基波频率波动对谐波分析的影响.通过对EAST实验数据的分析,验证了该谐波...     

Vibrational isotope effect by the low rank perturbation method

Mathematical formalism of the Low Rank Perturbation method (LRP) is applied to the vibrational isotope effect in the harmonic approximation with a standard assumption that force field does not change under isotopic substitutions. A pair of two n-atom isotopic molecules A and B which are identical except for isotopic substitutions at ρ atomic sites is considered. In the LRP approach vibrational frequencies ω _k and normal modes of the isotopomer B are expressed in terms of the vibrational frequencies ν _i and normal modes of the parent molecule A. In those relations complete specification of the normal modes is not required. Only amplitudes at sites τ affected by the isotopic substitutions and in the coordinate direction s (s = x, y, z) are needed. Out-of-plane vibrations of the (H,D)-benzene isotopomers are considered. Standard error of the LRP frequencies with respect to the DFT frequencies is on average . This error is due to the uncertainty of the input data (± 0.5 cm⁻¹) and in the absence of those uncertainties and in the harmonic approximation it should disappear. In comparing with experiment, one finds that LRP frequencies reproduces experimental frequencies of (H,D)-benzene isotopomers better () than scaled DFT frequencies () which are designed to minimize (by frequency scaling technique) this error. In addition, LRP is conceptually and numerically simple and it also provides a new insight in the vibrational isotope effect in the harmonic approximation.     

An Improved Plotting Package Applied to Chemical Graphing Problems

Abstract

A plotting package, written in LINC assembly language, is described. It produces high quality graphs and labels the graphs for direct submission to professional journals. A μ-LINC (Laboratory INstrument (Computer) is used to compute least-squares fitted curves and lines from data entered either on-or off-line and to draw the graphs directly on an incremental plotter. Bar graphs, line graphs, and fitted data can be plotted with options to suppress the lines and/or the points.