Graphs whose critical groups have larger rank

The critical group C(G) of a graph G is a refinement of the number of spanning trees of the graph and is closely connected with the Laplacian matrix. Let r(G) be the minimum number of generators (i.e., the rank) of the group C(G) and β(G) be the number of independent cycles of G. In this paper, some forbidden induced subgraphs are given for r(G) = n − 3 and all graphs with r(G) = β(G) = n − 3 are characterized.     

以(真空紫外光+紫外光)-共振加强式多光子游离法侦测交叉分子束之甲基生成物

用 (真空紫外光 +紫外光 ) 共振加强式多光子游离法侦测甲基自由基 .为证明此法之灵敏度 ,实验用交叉分子束反应F +CH4 (CD4 )→HF(DF) +CH3(CD3) ,其绝对灵敏度可达 10 7/cm3.由所得之光谱 ,更精确地得到两个里德堡态 3d2 E″及 3d2 A1′之光谱系数     

An inverse spectral problem

In this article an infinite periodic Jacobi matrix is under consideration. It is shown that the spectrum of the matrix consists of a single finite interval if and only if the period of the matrix is equal to unity.     

High performance liquid chromatographic analysis of six conjugated and unconjugated estrogens in serum.

Six estrogens-estrone, equilin, estradiol, sodium estrone sulphate, sodium equilin sulphate and sodium 17-alpha-dihydroequilin sulphate in dog serum were successfully separated and quantified by a high performance liquid chromatographic method developed in our laboratories. The mobile phase was optimized by studying the effects of an organic modifier (acetonitrile) and an ion pairing reagent (tetrabutylammonium hydroxide). The serum sample work-up procedure was designed to recover both conjugated and unconjugated estrogens. The optimum method involved acetonitrile to precipitate serum proteins/peptides and extract estrogens. Residues were reconstituted in 50% acetonitrile in water for injection. The detection limits for these six estrogens via UV detection ranged from 0.5 to 5 ng on-column with a signal-to-noise ratio (S/N) of 10 for a 20 microL injection and via fluorescence 0.1 ng on-column for 17-beta-estradiol. Validation data are included for all six estrogens.     

Planar graphs with maximum degree 4 are strongly 19-edge-colorable

A strong edge-coloring of a graph is a proper edge-coloring such that edges at distance at most 2 receive different colors. It is known that every planar graph has a strong edge-coloring by using at most $4\Delta +4$ colors, where $\Delta$ denotes the maximum degree of the graph. In this paper, we will show that 19 colors are enough to color a planar graph with maximum degree 4.     

On the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph

The Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph are the characteristic polynomials of its Laplacian matrix, signless Laplacian matrix and normalized Laplacian matrix, respectively. In this paper, we mainly derive six reduction procedures on the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph which can be used to construct larger Laplacian, signless Laplacian and normalized Laplacian cospectral graphs, respectively.     

On the second largest Laplacian eigenvalues of graphs

The second largest Laplacian eigenvalue of a graph is the second largest eigenvalue of the associated Laplacian matrix. In this paper, we study extremal graphs for the extremal values of the second largest Laplacian eigenvalue and the Laplacian separator of a connected graph, respectively. All simple connected graphs with second largest Laplacian eigenvalue at most 3 are characterized. It is also shown that graphs with second largest Laplacian eigenvalue at most 3 are determined by their Laplacian spectrum. Moreover, the graphs with maximum and the second maximum Laplacian separators among all connected graphs are determined.     

Improving artificial bee colony with one-position inheritance mechanism

Artificial bee colony (ABC) algorithm simulates the foraging behavior of honey bees. It shows good performance in many application problems and large scale optimization problems. However, variation of a solution in the ABC algorithm is only employed on one dimension of the solution. This would sometimes hamper the convergence speed of the ABC algorithm, especially for large scale optimization. This paper proposes a one-position inheritance (OPI) mechanism to overcome this drawback. The OPI mechanism aims to promote information exchange amongst employed bees of the ABC algorithm. For separable function, OPIABC has a higher probability resulting in function value improvement of the worst positions than ABC. Through one-position information exchange, the OPI mechanism can assist the ABC algorithm to find promising solutions. This mechanism has been tested on a set of 25 test functions with $D= 30$ and on CEC 2008 test suite with $D= 100$ and 1,000. Experimental results show that the OPI mechanism can speed up the convergence of the ABC algorithm. After the use of OPI, the performance of the ABC algorithm is significantly improved for both rotated problems and large scale problems. OPIABC is also competitive on both test suites comparing with other recently proposed swarm intelligence metaheuristics (e.g. SaDE and PSO2011). Furthermore, the OPI mechanism can greatly enhance the performance of other improved ABC algorithms.     

L（j,k）-number of Direct Product of Path and Cycle

For positive numbers j and k, an L(j,k)-labeling f of G is an assignment of numbers to vertices of G such that |f(u)-f(v)|≥j if uv∈E(G), and |f(u)-f(v)|≥k if d(u,v)=2. Then the span of f is the difference between the maximum and the minimum numbers assigned by f. The L(j,k)-number of G, denoted by λj,k(G), is the minimum span over all L(j,k)-labelings of G. In this paper, we give some results about the L(j,k)-number of the direct product of a path and a cycle for j≤k.     

Cyclic bandwidth with an edge added

B_c(G) denotes the cyclic bandwidth of graph G. In this paper, we obtain the maximum cyclic bandwidth of graphs of order p with adding an edge as follows: