Hamiltonian paths in polyhexes: The use of branching subgraphs to assist diagnosis of graph traceability

The traceability of some of the smaller polyhexes is examined. (A graph is said to be traceable or to have a Hamiltonian path if it has a path visiting every vertex just once.) Most polyhexes are traceable, and an attempt is made to develop some practical guidelines for finding those that are not. A subgraph consisting of the branching vertices of a polyhex, and of any edges which join pairs of such vertices, is a useful tool for this purpose. The principal resonance structures of such a graph suggest ways of finding simpler spannng subgraphs of the polyhex that will often make its traceability, or lack of it, more obvious.     

Hamiltonian circuits,Hamiltonian paths and branching graphs of benzenoid systems

A benzenoid systemH is a finite connected subgraph of the infinite hexagonal lattice with out cut bonds and non-hexagonal interior faces. The branching graphG ofH consists of all vertices ofH of degree 3 and bonds among them. In this paper, the following results are obtained:

1. A necessary condition for a benzenoid system to have a Hamiltonian circuit.
2. A necessary and sufficient condition for a benzenoid system to have a Hamiltonian path.
3. A characterization of connected subgraphs of the infinite hexagonal lattice which are branching graphs of benzenoid systems.
4. A proof that if a disconnected subgraph G of the infinite hexagonal lattice given along with the positions of its vertices is the branching graph of a benzenoid system H, then H is unique.

     

Weak aggregation: State of the art,expectations and open questions

This paper gives an overview of weak aggregation due to long-range molecular forces beyond the first neighbor. Such subtle self-assemblies are an important part of modern colloidal chemistry and concern organic molecules as well as inorganic electrolytes and hybrid aggregates. Diverse aspects of such colloidal aggregations, as described in this special issue, can be characterized by the effective free energy per molecule involved. We discuss here expectations about emerging knowledge in this field and predictive modeling of inorganic as well as organic colloids and hybrid aggregates. Some still open questions are also given.     

Graphical properties of polyhexes: Perfect matching vector and forcing

From the viewpoint of graph theory and its applications, subgraphs of the tiling of the plane with unit squares have long been studied in statistical mechanics, In organic chemistry, a much more relevant case concerns subgraphs of the tiling with unit hexagons. Our purpose here is to take a mathematical view of such polyhex graphsG and study two novel concepts concerning perfect matchingsM. First, the forcing number ofM is the smallest number of edges ofM which are not contained in any other perfect matching ofG. Second, the perfect matching vector ofM is written (n ₃,n ₂,n ₁,n ₀), wheren _k is the number of hexagons with exactlyk edges inM. We establish some initial results involving these two concepts and pose some questions.     

On the recognition of structural features: Some general problems illustrated by hamiltonian paths in polyhexes

Human and machine recognition skills are discussed, though not comprehensively reviewed, and some of the difficulties are illustrated by algorithms written to search for Hamiltonian paths in polyhexes. The most successful strategy for this is based upon the branching graph, a recently introduced graph-theoretical device which can aid the recognition of edges that arenot part of a Hamiltonian path. Another, more widely applicable approach that is interesting, although in this preliminary form only a little better than random methods, uses the metaphor of biological evolution, and tries to breed and grow paths subjected to natural selection.     

Computer generation of characteristic polynomials of edge-weighted graphs,heterographs, and directed graphs

The computer code developed previously (K. Balasubramanian, J. Computational Chem., 5 , 387 (1984)) for the characteristic polynomials of ordinary (nonweighted) graphs is extended in this investigation to edge-weighted graphs, heterographs (vertex-weighted), graphs with loops, directed graphs, and signed graphs. This extension leads to a number of important applications of this code to several areas such as chemical kinetics, statistical mechanics, quantum chemistry of polymers, and unsaturated systems containing heteroatoms which include bond alternation. The characteristic polynomials of several edgeweighted graphs which may represent conjugated systems with bond alternations, heterographs (molecules with heteroatoms), directed graphs (chemical reaction network), and signed graphs and lattices are obtained for the first time.     

The temperature-dependence of elementary reaction rates: beyond Arrhenius

The rates of chemical reactions and the dependence of their rate constants on temperature are of central importance in chemistry. Advances in the temperature-range and accuracy of kinetic measurements, principally inspired by the need to provide data for models of combustion, atmospheric, and astrophysical chemistry, show up the inadequacy of the venerable Arrhenius equation--at least, over wide ranges of temperature. This critical review will address the question of how to reach an understanding of the factors that control the rates of 'non-Arrhenius' reactions. It makes use of a number of recent kinetic measurements and shows how developments in advanced forms of transition state theory provide satisfactory explanations of complex kinetic behaviour (72 references).     

The polymorphism of ice: five unresolved questions

Our recent discovery of three new phases of ice has increased the total number of known distinct polymorphs of ice to fifteen. In this Perspective article, we give a brief account of previous work in the field, and discuss some of the particularly interesting open questions that have emerged from recent studies. These include (i) the effectiveness of acid and base dopants to enable hydrogen-ordering processes in the ices, (ii) the comparison of the calorimetric data of some of the crystalline phases of ice and low-density amorphous ice, (iii) the disagreement between the experimental ice XV structure and computational predictions, (iv) the incompleteness of some of the hydrogen order/disorder pairs and (v) the new frontiers at the high and negative pressure ends of the phase diagram.     

The branching point in the petasaccharides viridopentaoses A,B and C

The branching point in the viridopentaoses has been determined by a three step sequence, permethylation/methanolysis/benzoylation, followed by CD measurements, a method which can be scaled down to microgram quantities.     

Revisiting the foundations of the quantum theory of atoms in molecules: Some open problems

In a series of papers in the last 10 years, various aspects of the mathematical foundations of the quantum theory of atoms in molecules have been considered by this author and his coworkers in some details. Although these considerations answered part of the questions raised by some critics on the mathematical foundations of the quantum theory of atoms in molecules, however, new mathematical problems also emerged during these studies that were reviewed elsewhere [Sh. Shahbazian Int. J. Quantum Chem. 2011 , 111, 4497.]. Beyond mathematical subtleties of the formalism that were the original motivation for initial exchanges and disputes, the questions raised by critics had a constructive effect and prompted the author to propose a novel extension of the theory, now called the multi‐component quantum theory of atoms in molecules [M. Goli, Sh. Shahbazian Theor. Chem. Acc. 2013 , 132, 1365.]. Taking this background into account, in this paper a new set of open problems is put forward that the author believes proper answers to these questions, may open new doors for future theoretical developments of the quantum theory of atoms in molecules. Accordingly, rather than emphasizing on the rigorous mathematical formulation, the practical motivations behind proposing these questions are discussed in detail and the relevant literature are reviewed while when possible, evidence and routes to answers are also provided. The author hopes that proposing these open questions as a compact package may motivate more mathematically oriented people to participate in future developments of the quantum theory of atoms in molecules and its multi‐component version.