Variable Neighborhood Search.

The Wiener index W is the sum of distances between all pairs of vertices of a (connected) graph. The paper outlines the results known for W of trees: methods for computation of W and combinatorial expressions for W for various classes of trees, the isomorphism–discriminating power of W, connections between W and the center and centroid of a tree, as well as between W and the Laplacian eigenvalues, results on the Wiener indices of the line graphs of trees, on trees extremal w.r.t. W, and on integers which cannot be Wiener indices of trees. A few conjectures and open problems are mentioned, as well as the applications of W in chemistry, communication theory and elsewhere.

Wiener Index of Trees: Theory and Applications

The fact that a new text book introducing the essentials of computational chemistry contains more than 500 pages shows impressively the grown and still growing size and importance of this field of chemistry. The author’s objectives of the book, using his own words, are “to provide a survey of computational chemistry its underpinnings, its jargon, its strengths and weaknesses that will be accessible to both the experimental and theoretical communities”. This design as a general introduction into computational chemistry makes it an alternative to Andrew R. Leach’s well-established “Molecular Modeling” (Prentice Hall) and Frank Jensen’s “Introduction to Computational Chemistry” (Wiley), although the latter focuses on the theory of electronic structure methods. Cramer’s “Essentials” covers force field and molecular orbital theory, Monte Carlo and Molecular Dynamics simulations, thermodynamic and electronic (spectroscopic) property calculation, condensed phase treatment and a few more topics. Moreover, the book contains thirteen selected case studies sexamples taken from the literature sto illustrate the application of the just presented theoretical and computational models. This especially makes the text book well suited for both classroom discussion and self-study. Each chapter of “Essentials” covers a main topic of computational chemistry and will be briefly described here; all chapters are ended by a bibliography and suggested additional readings as well as the literature references cited in the text. In chapter 1 the author defines basic terms such as “theory”, “model”, and “computation”, introduces the concept of the potential energy surface and provides some general considerations about hardware and software. Interestingly, the first equation occurring in the text is not Schro ̈dinger’s equation, as is the case for most computational chemistry introductions, but the famous Einstein relation. The second chapter deals with molecular mechanics. It explains the different potential energy contributions, introduces the field of structure optimization, and provides an overview of the variety of modern force fields. Chapter 3 covers the simulation of molecular ensembles. It defines phase space and trajectories and shows the formalism of, and problems and difference between, Monte Carlo and molecular dynamics. In chapter 4 the author introduces the foundations of molecular orbital theory. Basic concepts such as Hamilton operator, LCAO basis set approach, many-electron wave functions, etc. are explained. To illuminate the LCAO variational process, the Hu ̈ckel theory is presented with an example. Chapter 5 deals with semiempirical molecular orbital (MO) theory. Besides the classical approaches (extended Hu ̈ckel, CNDO, INDO, NDDO) and methods (e.g., MNDO, AM1, PM3) and their performance, examples are provided from the ongoing development in that still fascinating area. Ab initio MO theory is presented in chapter 6; the basis set concept is discussed in detail, and, after some considerations from an user’s point of view, the general performance of ab initio methods is explicated. The next chapter covers the problem of electron correlation and gives the most prominent solutions for its treatment: configuration interaction, theory of the multiconfiguration self-consistent field, perturbation, and coupled cluster. Practical issues are also discussed. Chapter 8’s topic is density functional theory (DFT). Its theoretical foundation, methodology, and some functionals as well as its pros and cons compared to MO theory are presented together with a general performance overview. The next two chapters deal with charge distribution, derived and spectroscopic properties (e.g., atomic charges, polarizability, rotational, vibrational, and NMR spectra), and thermodynamic properties (e.g., zero-point vibrational energy, free energy of formation, and reaction). The modeling of condensed phases is addressed in chapters 11 (implicit models) and 12 (explicit models), which closes with a comparison between the two approaches. Chapter 13 familiarizes the reader with hybrid quantum mechanical/molecular mechanical (QM/MM) models. Polarization as well as the problematic implications of unsaturated QM and MM components are discussed, and empirical valence bond methods are also presented. The treatment of excited states is the topic of chapter 14; besides CI and MCSCF as computational methods, transition probabilities and solvatochromism are discussed. The last chapter deals with reaction dynamics, mostly adiabaticskinetics, rate constants, reaction paths, and transition state theory are section topics here sbut also nonadiabatic, introducing curve crossing and Marcus theory in brief. The appendix is divided into four parts: an acronym glossary (which is very helpful), an overview of symmetry and group theory, an introduction to spin algebra, and finally a section about orbital localization. A rather detailed index ends the book. The “Essentials” writing style fits the fascinating topic: one reads on and on andssurprise! sanother chapter has been absorbed. The text is complemented by a large number of black and white figures and clear tables, mostly self-explanatory with descriptive captions. The use of equations and mathematical formulas in general is well-balanced, and the level of math should be understandable for every natural scientist with some basic knowledge of physics. There are only a few minor shortcomings: for example, a literature reference cited in the text (“Beck et al.”, p 142) is missing in the bibliography; “Kronecker” is mistyped with o ̈; and the author completely forgot to reference Leach’s text book when referring to other computational chemistry introductions. However, the author has established a specific errata web page (http://pollux.chem.umn.edu/ ∼cramer/Errors.html) with all known errors. These will be corrected in the next printing or next revised edition, respectively. With its emphasis, on one hand, on the basic concepts and applications rather than pure theory and mathematics, and on the other hand, coverage of quantum mechanical and classical mechanical models including examples from inorganic, organic, and biological chemistry, “Essentials” is a useful tool not only for teaching and learning but also as a quick reference, and thus will most probably become one of the standard text books for computational chemistry.

Essentials Of Computational Chemistry Theories And Models

Modern Graph Theory

Variable neighbourhood search (VNS) is a metaheuristic, or a framework for building heuristics, based upon systematic changes of neighbourhoods both in descent phase, to find a local minimum, and in perturbation phase to emerge from the corresponding valley It was first proposed in 1997 and has since then rapidly developed both in its methods and its applications In the present paper, these two aspects are thoroughly reviewed and an extensive bibliography is provided Moreover, one section is devoted to newcomers It consists of steps for developing a heuristic for any particular problem Those steps are common to the implementation of other metaheuristics

Variable neighbourhood search: methods and applications

The Wiener index, W , is the sum of distances between all pairs of vertices in a graph G. The quadratic line graph is defined as L(L(G)), where L(G) is the line graph of G. A generalized star S is a tree consisting of ∆ ≥ 3 paths with the unique common endvertex. A relation between the Wiener index of S and of its quadratic graph is presented. It is shown that generalized stars having the property W (S) = W (L(L(S)) exist only for 4 ≤ ∆ ≤ 6. Infinite families of generalized stars with this property are constructed.

http://www.discuss.wmie.uz.zgora.pl/php/discuss3.php?ip=&url=pdf&nIdA=14896

Wiener index of generalized stars and their quadratic line graphs

Graphs of unbranched hexagonal systems consist of hexagonal rings connected with each other. Molecular graphs of unbranched polycyclic aromatic hydrocarbons serve as an example of graphs of this class. The Wiener index (or the Wiener number) of a graph is defined as the sum of distances between all pairs of its vertices. Necessary conditions for the existence of graphs with different numbers of hexagonal rings and equal values of the Wiener index are formulated, and examples of such graphs are presented.

Graphs of unbranched hexagonal systems with equal values of the Wiener index and different numbers of rings

Wiener Index of Hexagonal Chains with Segments of Equal Length

New matrices associated with graphs and induced global and local topological indices of molecular graphs were proposed recently by Diudea, Minailiuc and Balaban. These matrices in canonical form are matrix graph invariants. A combined degeneracy of such invariants is considered. For every case of degeneracy corresponding graphs are presented.

Degeneracy of some matrix graph invariants

An explicit relation between two well-known polynomial invariants of spatial graphs is established If g is a diagram of a spatial graph with p vertices and q edges, then Y (g; t) = (−t− t−1)p− q R (g ′; t), where Y (g) is the Yoshinaga polynomial of g and R(g ′) is the Yamada polynomial of its mirror image g ′

Andrey A. Dobrynin

Papers

Wiener index of generalized stars and their quadratic line graphs

Graphs of unbranched hexagonal systems with equal values of the Wiener index and different numbers of rings

Wiener Index of Hexagonal Chains with Segments of Equal Length

Degeneracy of some matrix graph invariants

On the Yoshinaga polynomial of spatial graphs