Study of fullerenes via their symmetry groups

In this paper, we compute the automorphism group of some infinite classes of fullerene graphs and then we compute their symmetric Szeged index which is a new topological index based on the automorphism group.     

交错群A_5的4度连通对称图的分类

一个图Г称为G-对称的(symmetric).如果其同构群Aut(r)的一个子群G在图r的有向孤集(set of ordered pairs of adjacent vertices)上的作用是传递的(transitive).本文主要结果是:设图Г是4度对称图.全自同构群Aut(r)=A_5,则图r是且仅是如下图之一:(1)Г是15个点的完全图K_5的三维覆盖(3-fold cover)图.(2)Г是完全图K_5.     

New Isomerization Mechanisms for Fullerenes

Abstract

There are proposed two classes of isomerization transformations for fullerene graphs: rotation and mirror reflection of the chosen part of the graph. An algorithm for generation of extended isomerization maps is developed. The algorithm is applied to C₄₀ fullerene as an example.     

Multiple Decision Procedures: Theory and Methodology of Selecting and Ranking Populations

In planning a fractional factorial experiment prior knowledge may suggest that some interactions are potentially important and should therefore be estimated free of the main effects. In this article, we propose a graph-aided method to solve this problem for two-level experiments. First, we choose the defining relations for a 2 ^n–k design according to a goodness criterion such as the minimum aberration criterion. Then we construct all of the nonisomorphic graphs that represent the solutions to the problem of simultaneous estimation of main effects and two-factor interactions for the given defining relations. In each graph a vertex represents a factor and an edge represents the interaction between the two factors. For the experiment planner, the job is simple: Draw a graph representing the specified interactions and compare it with the list of graphs obtained previously. Our approach is a substantial improvement over Taguchi's linear graphs.     

An efficient method for decomposition of regular structures using graph products

In this paper an efficient method is presented for calculating the eigenvalues of regular structural models. A structural model is called regular if they can be viewed as the direct or strong Cartesian product of some simple graphs known as their generators. The eigenvalues of the adjacency and Laplacian matrices for a regular graph model are easily obtained by the evaluation of eigenvalues of its generators. The second eigenvalue of the Laplacian of a graph is also obtained using a much faster and much simple approach than the existing methods. Copyright © 2004 John Wiley & Sons, Ltd.     

Hyperelliptic curves with reduced automorphism group A 5

We study genus g hyperelliptic curves with reduced automorphism group A ₅ and give equations y ² = f(x) for such curves in both cases where f(x) is a decomposable polynomial in x ² or x ⁵. For any fixed genus the locus of such curves is a rational variety. We show that for every point in this locus the field of moduli is a field of definition. Moreover, there exists a rational model y ² = F(x) or y ² = x F(x) of the curve over its field of moduli where F(x) can be chosen to be decomposable in x ² or x ⁵. While similar equations have been given in (Bujalance et al. in Mm. Soc. Math. Fr. No. 86, 2001) over , this is the first time that these equations are given over the field of moduli of the curve.     

C68: A non-IPR fullerene capable of binding extraordinary amounts of Cs atoms

The non-IPR fulleride compound Cs_xC₆₈ was investigated in analogy to previous experiments on Cs_xC₅₈. While the vibrational properties as well as the electronic structure in the valence band range appear to be pretty similar to Cs_xC₅₈, a striking difference could be observed by XPS and Photoemission Microscopy analysis: apparently, C₆₈ forms different phases within a thick film including islands in which it is able to store more than 70 Cs atoms per fullerene cage and to keep these amounts even after annealing over 1100 K. This means that Cs_xC₆₈ constitutes the highest alkali metal doping degree of any Fullerene species known so far. Spatially resolved photoemission analysis also revealed that the Cs-rich fulleride phases constitute a unique chemical state that has never been observed before.     

Block diagonalization of Laplacian matrices of symmetric graphs via group theory

In this article, group theory is employed for block diagonalization of Laplacian matrices of symmetric graphs. The inter‐relation between group diagonalization methods and algebraic‐graph methods developed in recent years are established. Efficient methods are presented for calculating the eigenvalues and eigenvectors of matrices having canonical patterns. This is achieved by using concepts from group theory, linear algebra, and graph theory. These methods, which can be viewed as extensions to the previously developed approaches, are illustrated by applying to the eigensolution of the Laplacian matrices of symmetric graphs. The methods of this paper can be applied to combinatorial optimization problems such as nodal and element ordering and graph partitioning by calculating the second eigenvalue for the Laplacian matrices of the models and the formation of their Fiedler vectors. Considering the graphs as the topological models of skeletal structures, the present methods become applicable to the calculation of the buckling loads and the natural frequencies and natural modes of skeletal structures. Copyright © 2006 John Wiley & Sons, Ltd.     

The block-transitive,point-imprimitive 2-(729, 8, 1) designs

The block-transitive point-imprimitive 2-(729,8,1) designs are classified. They all have full automorphism group of order 729.13 which is an extension of a groupN of order 729, acting regularly on points, by a group of order 13. There are, up to isomorphism, 27 designs withN elementary abelian, 13 designs withN=Z ₉ ³ and 427 designs withN the relatively free 3-generator, exponent 3, nilpotency class 2 group, a total of 467 designs. This classification completes the classification of block-transitive, point-imprimitive 2-(, k, 1) designs satisfying , which is the Delandtsheer-Doyen upper bound for the number of points of such designs. The only examples of block-transitive, point-imprimitive 2-(, k, 1) designs with are the 2-(729, 8, 1) designs constructed in this paper.The first three authors acknowledge the support of an Australian European Awards Program scholarship, a Deutsche Akademische Austauschdienst scholarship, and an Australian Research Council Research Fellowship, respectivelyThe authors wish to thank Brendan McKay for his independent verification of the non-isomorphism of the classes of designs found, and of their automorphism groups, using different, nauty techniques [6].     

THE SINGLE CRYSTAL X-RAY STRUCTURE OF C70Me8; THE ORIGIN OF THE FULLERENE ADDITION PATTERNS FOR BULKY GROUPS

ABSTRACT

The first complete proof of the equatorial addition pattern in [70]fullerene is provided by a single crystal x-ray structure of C₇₀Me₈. Compared to the bonds in the parent fullerene, those having a methyl group attached are lengthened, those having two methyl groups adjacent at each end are shortened due to compression by these groups, and some hexagonal rings show greater delocalisation (more aromaticity) than the corresponding rings in the precursor fullerene. In contrast to the 1,2-addition of sterically undemanding groups, bulky groups generally undergo 1,4-addition, but only give 1,2-addition if there is a need to quench an intermediate radical. The manner in which this gives rise to the patterns seen in C₆₀ X ₆, C₆₀ X ₈ and C₇₀ X ₁₀, where X is bulky, is proposed.     

Topological Characterization of Book Graph and Stacked Book Graph

Degree based topological indices are being widely used in computer-aided modeling, structural activity relations, and drug designing to predict the underlying topological properties of networks and graphs. In this work, we compute the certain important degree based topological indices like Randic index, sum connectivity index, ABC index, ABC₄ index, GA index and GA₅ index of Book graph B_n and Stacked book graph B_m,n. The results are analyzed by using edge partition, and the general formulas are derived for the above-mentioned families of graphs.     

Graph coloration and group theory for factorization of symmetric dynamic systems

Summary In this article the concept of graph coloration for factorization of the characteristic polynomial of a weighted directed graph is introduced and using the fundamental lemma of Petersdorf and Sachs, it is shown that this concept can be extended to other spectra of graph among which is the spectrum of the Laplacian matrix of graphs. The goal is to use this concept for factorizing the graphs having different types of symmetry properties. In order to present an efficient algorithm, some concepts from group theory are needed. The combination of group theory and graph coloration provides a powerful method for factorizing various structural matrices, an example of which is the stiffness matrices of mass-spring dynamic systems.     

Diphenylmethane Fulleroid C73H10: AM1 Computational Study

Diphenylmethane fulleroid C₇₃H₁₀ is studied by means of the AM1 quantum-chemical method. Full geometry optimization of its C _2v structure is performed together with vibrational harmonic analysis. The computed length of the C-C bond between two cage-hexagons bridged by the tetra-hedral C atom is 1.565 Å. Vibrational frequencies with a substantial IR intensity are reported.     

The New Method for the Synthesis of Fullerols Based on Radical Reaction

Abstract

The method for the synthesis of fullerols C₆₀(OH) _x was developed, allowing to obtain products with low number of hydroxy groups attached to fullerene molecule. At the first stage, product C₆₀(tert‐BuO) _x was obtained by the radical reaction of C₆₀ with tert‐butyl‐peroxide under heating or UV‐irradiation. At the second stage, fullerol C₆₀(OH) _x was synthesized after the elimination of tert‐butyl groups by acid treatment. Several samples C₆₀(OH) _x containing different numbers (x = 2–5) of hydroxy groups were obtained, the main product had x = 2. The structure of the fullerols was confirmed by UV‐ and Fourier‐transform infrared (FTIR)‐spectroscopy and MALDI‐TOF mass‐spectrometry.     

反拟阵自同构群的公理系统及其性质

给出了反拟阵自同构群的定义,利用反拟阵的公理系统建立其自同构群公理系统;在此基础上,讨论了拟阵与反反拟阵之间的一些关系。     

New Polymeric Phase in Low‐Doped Lithium Intercalated Fullerides

Abstract

The low‐doped Li _x C₆₀ compounds (x≤6) were investigated using laboratory X‐ray and synchrotron radiation diffraction, ¹³C NMR and Raman spectroscopy. Li₄C₆₀ shows an unusual 2D polymerisation in which the C₆₀ units are connected both by [2+2] cycloaddition and by single carbon–carbon bonds, a unique feature among the known polymerised fullerene compounds. This picture is fully supported also by static NMR and Raman measurements. The charge transfer to C₆₀ in the polymeric phase was evaluated from the shift of the A_g(2) mode. The depolymerisation process was investigated as well; despite the presence of two different bonds, the polymer‐to‐monomer transition induced by thermal treatments is a single‐step phenomenon.     

Facile and scalable synthesis of a highly hydroxylated water-soluble fullerenol as a single nanoparticle

A water-soluble polyhydroxylated fullerene, i.e. a fullerenol, with 44 hydroxyl groups and 8 secondary bound water molecules, C₆₀(OH)₄₄·8H₂O (estimated average structure), has been synthesized in a facile one step reaction from pristine C₆₀ by hydroxylation with hydrogen peroxide in the presence of a phase-transfer catalyst, tetra-n-butylammonium hydroxide (TBAH), under organic/aqueous bilayer conditions. The fullerenol exhibited high water solubility, up to 64.9 mg/mL, under neutral (pH = 7) conditions. Dynamic light-scattering (DLS) analysis showed a narrow particle size distribution, of 1–2 nm, indicating that the fullerenol had high dispersion properties in water. The results of particle size analyses, which both focused on a single nanoregion and were conducted using a novel induced grating (IG) method and a scanning probe microscope (SPM), were consistent with the DLS results. A plausible reaction mechanism, which includes fullerene oxide intermediates detected by liquid chromatography-mass spectrometry (LC-MS), has been proposed.     

An approach to machining process optimization

Optimization is rightly claimed to be the most significant factor distinguishing the modern approach to machining processes from the orthodox one. Practical results can be obtained from the application of mathematical modelling and optimization techniques.

The paper presents a procedure for solution of machining process optimization problems, with the use of theory of graphs and Bellman's optimum principle.

The finite graph G=(N, A) (where -N denoted a set of nodes and A the set of arcs) here presents the set of feasible solutions. Each are represents a corresponding machining operation and each path the feasible machining process.

A number can be associated with each are : the increment of criterion for optimality. Using the optimum principle we can find the shortest possible path in the graph, thus solving given problems of choice for the optimal machining process.

There is a variety of ways in which the performance criterion can be formulated. In this paper only those performance measures having sufficient economical justification are considered. We may minimize the costs of machining processes, or optimize economical effectiveness of capital engaged.

The method presented can be applied with every additive criterion, as well as with some non-additive ones. A practical example illustrates the field of possible application of this method.     

Node-pair reliability of network systems with small distances between adjacent nodes

A new method for computing the node-pair reliability of network systems modeled by random graphs with nodes arranged in sequence is presented. It is based on a recursive algorithm using the “sliding window” technique, the window being composed of several consecutive nodes. In a single step, the connectivity probabilities for all nodes included in the window are found. Subsequently, the window is moved one node forward. This process is repeated until, in the last step, the window reaches the terminal node. The connectivity probabilities found at that point are used to compute the node-pair reliability of the network system considered. The algorithm is designed especially for graphs with small distances between adjacent nodes, where the distance between two nodes is defined as the absolute value of the difference between the nodes’ numbers. The maximal distance between any two adjacent nodes is denoted by Γ(G), where G symbolizes a random graph. If Γ(G)=2 then the method can be applied for directed as well as undirected graphs whose nodes and edges are subject to failure. This is important in view of the fact that many algorithms computing network reliability are designed for graphs with failure-prone edges and reliable nodes. If Γ(G)=3 then the method's applicability is limited to undirected graphs with reliable nodes. The main asset of the presented algorithms is their low numerical complexity—O(n), where n denotes the number of nodes.