On the pagenumber of trivalent Cayley graphs

Book embedding of graphs is one of the graph layout problem. It is useful for the multiprocessor network layout or the fault-tolerant processor arrays. We show that the trivalent Cayley graphs proposed by Vadapalli and Srimani can be embedded in five pages, and show some additional results on cube-connected cycles.     

具有唯一4度最大度点的图的全色数

一个图Ｇ的全色数χ_T（Ｇ）是使得Ｖ（Ｇ）∪Ｅ（Ｇ）中相邻或相关联元素均染不同颜色的最少颜色数．文中证明了，若图Ｇ只有唯一的一个４度最大度点，则χ_T（Ｇ）＝Δ（Ｇ）＋１．     

Functional Central Limit Theorems for Triangular Arrays of Function-Indexed Processes under Uniformly Integrable Entropy Conditions

Functional central limit theorems for triangular arrays of rowwise independent stochastic processes are established by a method replacing tail probabilities by expectations throughout. The main tool is a maximal inequality based on a preliminary version proved by P. Gaenssler and Th. Schlumprecht. Its essential refinement used here is achieved by an additional inequality due to M. Ledoux and M. Talagrand. The entropy condition emerging in our theorems was introduced by K. S. Alexander, whose functional central limit theorem for so-calledmeasure-like processeswill be also regained. Applications concern, in particular, so-calledrandom measure processeswhich include function-indexed empirical processes and partial-sum processes (with random or fixed locations). In this context, we obtain generalizations of results due to K. S. Alexander, M. A. Arcones, P. Gaenssler, and K. Ziegler. Further examples include nonparametric regression and intensity estimation for spatial Poisson processes.     

Diagrams,orientations, and varieties

One of the central problems in the theory of ordered sets is to describe the orientations of the covering graph of an ordered set. We show that the particular operation called inversion, together with the classical constructions of retraction and product, provide a context for the classification of all such orientations. AMS subject classifications (1980). 06A10, 05C75, 05C20.     

The dielectric properties of water in its different states of interaction

A tutorial on dielectric (relaxation) spectrometry of liquids is given in this article. Some methods of measuring complex (electric) permittivity spectra are briefly described. Results for water are presented and related to characteristic properties of the liquid structure and to models of the molecular dynamics, particularly as resulting from computer simulation studies. Dielectric spectra for aqueous solutions of low weight electrolytes, polyelectrolytes, small molecules, and polymers are discussed to illustrate effects of kinetic depolarization, structure saturation, as well as positive, negative, and hydrophobic hydration. Reference is also made to fluctuations in the hydrogen bond network of mixtures of water with liquids that are completely miscible with this unique solvent.     

Global Forcing Number of Benzenoid Graphs

A global forcing set in a simple connected graph G with a perfect matching is any subset S of E(G) such that the restriction of the characteristic function of perfect matchings of G on S is an injection. The number of edges in a global forcing set of the smallest cardinality is called the global forcing number of G. In this paper we prove several results concerning global forcing sets and numbers of benzenoid graphs. In particular, we prove that all catacondensed benzenoids and catafused coronoids with n hexagons have the global forcing number equal to n, and that for pericondensed benzenoids the global forcing number is always strictly smaller than the number of hexagons.     

Unicyclic Graphs with Minimal Energy

If G is a graph and ₁,₂,..., _n are its eigenvalues, then the energy of G is defined as E(G)=|₁|+|₂|++| _n |. Let S _n ³ be the graph obtained from the star graph with n vertices by adding an edge. In this paper we prove that S _n ³ is the unique minimal energy graph among all unicyclic graphs with n vertices (n6).     

The de novo design of median molecules within a property range of interest

Summary In this paper an application is presented of the median molecule workflow to the de novo design of novel molecular entities with a property profile of interest. Median molecules are structures that are optimised to be similar to a set of existing molecules of interest as an approach for lead exploration and hopping. An overview of this workflow is provided together with an example of an instance using the similarity to camphor and menthol as objectives. The methodology of the experiments is defined and the workflow is applied to designing novel molecules for two physical property datasets: mean molecular polarisability and aqueous solubility. This paper concludes with a discussion of the characteristics of this method.     

On the Permanental Polynomials of Some Graphs

Let G be a simple graph with adjacency matrix A(G) and (G,x) the permanental polynomial of G. Let G × H denotes the Cartesian product of graphs G and H. Inspired by Kleins idea to compute the permanent of some matrices (Mol. Phy. 31 (3) (1976) 811–823), in this paper in terms of some orientation of graphs we study the permanental polynomial of a type of graphs. Here are some of our main results.1.If G is a bipartite graph containing no subgraph which is an even subdivision of K _2,3, then G has an orientation G ^e such that (G,x) = det (xI-A(G ^e )), where A(G ^e ) denotes the skew adjacency matrix of G ^e.2.Let G be a 2-connected outerplanar bipartite graph with n vertices. Then there exists a 2-connected outerplanar bipartite graph with 2n+2 vertices such that (G,x) is a factor of .3.Let T be an arbitrary tree with n vertices. Then , where ₁ , ₂ , ..., _n are the eigenvalues of T.     

Pattern of separatrices and intrinsic reaction coordinates for degenerate thermal rearrangements

A structurally stable model of the standard adiabatic gradient field of the potential energy surface for certain pericyclic reactions is derived.These reactions are not subjected to the principles of orbital isomerism or to the Woodward-Hoffmann rules.Use is made of a principle established by Ariel Fernández and Oktay Sinanolu which precludes direct meta-IRC connections between transition states.It is shown that Jahn-Teller isomers of the singlet biradicals involved in the process are not interconvertible since the biradical configuration is not a transition state but a critical point with Hessian matrix with two negative eigenvalues.The topological features of the PES obtained by combinatorial methods are in full agreement with earlier results obtained from MINDO calculations.