On the centrality of vertices of molecular graphs

For acyclic systems the center of a graph has been known to be either a single vertex of two adjacent vertices, that is, an edge. It has not been quite clear how to extend the concept of graph center to polycyclic systems. Several approaches to the graph center of molecular graphs of polycyclic graphs have been proposed in the literature. In most cases alternative approaches, however, while being apparently equally plausible, gave the same results for many molecules, but occasionally they differ in their characterization of molecular center. In order to reduce the number of vertices that would qualify as forming the center of the graph, a hierarchy of rules have been considered in the search for graph centers. We reconsidered the problem of “the center of a graph” by using a novel concept of graph theory, the vertex “weights,” defined by counting the number of pairs of vertices at the same distance from the vertex considered. This approach gives often the same results for graph centers of acyclic graphs as the standard definition of graph center based on vertex eccentricities. However, in some cases when two nonequivalent vertices have been found as graph center, the novel approach can discriminate between the two. The same approach applies to cyclic graphs without additional rules to locate the vertex or vertices forming the center of polycyclic graphs, vertices referred to as central vertices of a graph. In addition, the novel vertex “weights,” in the case of acyclic, cyclic, and polycyclic graphs can be interpreted as vertex centralities, a measure for how close or distant vertices are from the center or central vertices of the graph. Besides illustrating the centralities of a number of smaller polycyclic graphs, we also report on several acyclic graphs showing the same centrality values of their vertices. © 2013 Wiley Periodicals, Inc.     

Random walks and their diagnostic value for characterization of atomic environment

A characterization of atomic environments based on counting random walks in a molecular skeleton is outlined. To each atom in a molecule a sequence of integers w₁, w₂, w₃,…, w_n is assigned, where w_i represents the number of self-returning walks of length k, the length being defined by the number of bonds traversed. Properties of the derived atom codes are discussed. The codes display an impressive diversity and are superior to atomic codes based on enumeration of self-avoiding walks (or paths) in discriminating atomic environments. In certain cases the codes of individual atoms are not unique and the same codes appear in different molecules or even within the same molecule. The occurrence of the nonunique codes can be related to special structural situations, associated with the occurrence of isospectral graphs. These isospectral graphs which have atoms with identical codes can generate additional isospectral structures by attaching any arbitrary group to such points. If nonequivalent atoms of a single molecule have identical random walk codes, substitution at the singular points alternatively will produce isospectral graphs. Examples of such situations are given.     

On walks in molecular graphs.

Walks in molecular graphs and their counts for a long time have found applications in theoretical chemistry. These are based on the fact that the (i, j)-entry of the kth power of the adjacency matrix is equal to the number of walks starting at vertex i, ending at vertex j, and having length k. In recent papers (refs 13, 18, 19) the numbers of all walks of length k, called molecular walk counts, mwc(k), and their sum from k = 1 to k = n - 1, called total walk count, twc, were proposed as quantities suitable for QSPR studies and capable of measuring the complexity of organic molecules. We now establish a few general properties of mwc's and twc among which are the linear dependence between the mwc's and linear correlations between the mwc's and twc, the spectral decomposition of mwc's, and various connections between the walk counts and the eigenvalues and eigenvectors of the molecular graph. We also characterize the graphs possessing minimal and maximal walk counts.     

Novel shape descriptors for molecular graphs.

We report on novel graph theoretical indices which are sensitive to the shapes of molecular graphs. In contrast to the Kier's kappa shape indices which were based on a comparison of a molecular graph with graphs representing the extreme shapes, the linear graph and the "star" graph, the new shape indices are obtained by considering for all atoms the number of paths and the number of walks within a graph and then making the quotients of the number of paths and the number of walks the same length. The new shape indices show much higher discrimination among isomers when compared to the kappa shape indices. We report the new shape indices for smaller alkanes and several cyclic structures and illustrate their use in structure-property correlations. The new indices offer regressions of high quality for diverse physicochemical properties of octanes. They also have lead to a novel classification of physicochemical properties of alkanes.     

To What Extent are “Atoms in Molecules” Structures of Hydrocarbons Reproducible from the Promolecule Electron Densities?

The “atoms in molecules” structures of 225 unsubstituted hydrocarbons are derived from both the optimized and the promolecule electron densities. A comparative analysis demonstrates that the molecular graphs derived from these two types of electron densities at the same geometry are equivalent for almost 90 % of the hydrocarbons containing the same number and types of critical points. For the remaining 10 % of molecules, it is demonstrated that by inducing small perturbations, through the variation of the used basis set or slight changes in the used geometry, the emerging molecular graphs from both densities are also equivalent. Interestingly, the (3, ?1) critical point between two “non‐bonded” hydrogen atoms, which triggered “H?H bonding” controversy is also observed in the promolecule densities of certain hydrocarbons. Evidently, the topology of the electron density is not dictated by chemical bonds or strong interactions and deformations induced by the interactions of atoms in molecules have a quite marginal role, virtually null, in shaping the general traits of the topology of molecular electron densities of the studied hydrocarbons, whereas the key factor is the underlying atomic densities.     

Study of $${\text{CD}}_{5}^{ + }$$ Ions and Deuterated Variants( $${\text{C}}{{{\text{H}}}_{x}}{\text{D}}_{{(5 - x)}}^{ + }$$ ): An Artefactual Rotation

Russian Journal of Physical Chemistry A - Deuterated methane, $${\text{CD}}_{5}^{ + }$$ , has unusual vibrational and rotational behavior because its three nonequivalent equilibrium structures have...     

Carbon nanotubes,science and technology part (I) structure,synthesis and characterisation

Since their discovery in 1991 by the Japanese scientist “Sumio Iijima”, carbon nanotubes have been of great interest, both from a fundamental point of view and for future applications. Different types of carbon nanotubes can be produced in various ways. Economically feasible large-scale production and purification techniques are still under development. Carbon nanotubes are discussed in this review in terms of history, types, structure, synthesis and characterisation methods. Carbon nanotubes have attracted the fancy of many scientists worldwide. The unique and unusual properties of these structures make them a unique material with a whole range of promising applications.     

A representation of the McClelland method of graph splitting. Stack graphs

McClelland's rules on graph splitting can be represented using the generalized graph notation. Generalized graphs are edge- and vertex-weighted graphs, which are becoming important to chemical problems. By this the McClelland method of graph splitting has a wider range of applications. “Stack graphs” are constructed from identical “base graphs” by connecting corresponding vertices from one base to another. Their eigenvalues are related to the eigenvalues of the base graph. Two- and even three-layered graphs may be used as a simple model for the inter-ring interaction in a cyclophane.     

The Adsorption of Ag on (CdTe)<Subscript>13</Subscript> Core-Cage Nanocluster: A Computational Study

Cadmium chalcogenide semiconductor quantum dots, especially doped nanoclusters, have attracted great attention for their effects on photo generated carriers and their lifetime due to introduced trapping states by changing surface unbonded orbitals. Here, we investigate the adsorption of Ag on “magic-sized” cadmium chalcogenide (CdTe)₁₃ core-cage nanoclusters, Cd₁₃Te₁₃Ag, by first-principles density functional theory. All possible adsorption sites, top, bridge, and hollow sites, have been considered. Particular attention is paid to the energy band structures of Cd₁₃Te₁₃Ag. The study demonstrates that the hollow sites, the centers of hexagons, are the favorite Ag adsorption sites. Unlike observed shallow acceptor level of doped QDs, two unusual deep mid-gap states with different spins, spin up and spin down, are observed. These two deep states shift with Ag moving towards the core of cage. The detail properties of adsorption configurations and these two deep states are analyzed. These two deep states should have important role to their optical applications.     

Synthesis of Biological Activities of Phosphorus-Containing Polycationic Strings

Abstract

Several series of phosphoruscontaining polycationic strings have been prepared. These include those in which quaternary phosphonium sites are located at regular intervals along a linear chain of defined length (“Strings”), and those in which the cationic phosphonium, sites are in linkages arrayed in branching arms about a cenhal focus unit (“baloons”). Polyphosphonium “balloon” arrays have been attached to a polystyrene backbone prowding a mataial which serves as an anionic exchange resin. PolyCationic strings have been prepared in which phosphorus is present a3 phosphine oxide functidties and the cationic sites are quatanary ammonium sites anayed at regular intervals along a linear chain of defined length. Several of the resultant materials have demonshated sigruficant antibaaaial activity. Interactions of the polycationic species with the anionic gmove region of double stranded DNA have been investigated.     

Computer enumeration of walks on directed graphs

A vectorized computer code is developed for the enumeration of walks through the matrix power method for directed graphs. Application of this code to several graphs is considered. It is shown that the coefficients in the generating functions for signed graphs are much smaller in magnitude. It is shown that self-avoiding walks on some graphs can be enumerated as a linear combination of walk GFs of directed paths and rooted-directed paths.     

Membrane‐Length Amphiphiles Exhibiting Structural Simplicity and Ion Channel Activity

A number of synthetic ion channels have been reported in recent years that incorporate unusual or sophisticated design elements. The present work demonstrates that extremely simple compounds can function as ion channels (insert in bilayers, exhibit open–close behavior) if they meet minimum criteria. A simple membrane spanning structure may function as a channel if 1) it possesses polar headgroups (is bolaamphiphilic), 2) possesses a “central relay,” and 3) channel function (open–close behavior) must be detected after insertion of the amphiphile directly into the aqueous liposomal or cellular suspension. We show here compounds that are simple spans to which we have given the name “aplosspan” (from the Greek απλoσ + span) that meet these criteria. They are similar to, but simpler than, structures reported in the literatures that incorporate more complex design features.     

Discrimination and ordering of chemical structures by the number of walks

The use of the number of walks for the discrimination of graphs representing chemical structures is discussed. A highly selective graph-theoretical index based on the number of walks is defined. The index values were calculated for 661 acyclic and 376 cyclic structures. The selectivity of the new index is compared to that of some of the most selective previously defined indexes. The ordering of structures induced by the value of the index is also considered.Presented in part at the 7th International Conference on Computers in Chemical Research and Education, held in Garmisch-Partenkirchen, Germany, June 10–14, 1985     

Molecular orbital theory of the hydrogen bond. 24. Ground-state water–uracil complexes

Ab initio SCF calculations with the STO -3G basis set have been performed to investigate the structural, energetic, and electronic properties of mixed water–uracil dimers formed at the six hydrogen-bonding sites in the uracil molecular plane. Hydrogen-bond formation at three of the carbonyl oxygen sites leads to cyclic structures in which a water molecule bridges N₁? H and O₂, N₃? H and O₂, and N₃? H and O₄. Open structures form at O₄, N₁? H, and N₃? H. The two most stable structures, with energies of 9.9 and 9.7 kcal/mole, respectively, are the open structure at N₁? H and the cyclic one at N₁? H and O₂. These two are easily interconverted, and may be regarded as corresponding to just one “wobble” dimer. At 1 kcal/mole higher in energy is another “wobble” dimer consisting of an open structure at N₃? H and a cyclic structure at N₃? H and O₄. The third cyclic structure at N₃? H and O₂ collapses to the “wobble” dimer at N₃? H and O₄. The two “wobble” dimers are significantly more stable than the open dimer formed at O₄, which has a stabilization energy of 5.4 kcal/mole. Uracil is a stronger proton donor to water through N₁? H than N₃? H, owing to a more favorable molecular dipole moment alignment when association occurs through H₁. Hydration of uracil by additional water molecules has also been investigated. Dimer stabilization energies and hydrogen-bond energies are nearly additive in most 2:1 water:uracil structures. There are three stable “wobble” trimers, which have stabilization energies that vary from 7 to 9 kcal/mole per water molecule. Hydrogen-bond strengths are slightly enhanced in 3:1 water:uracil structures, but the cooperative effect in hydrogen bonding is still relatively small. The single stable water–uracil tetramer is a “wobble” tetramer, with two water molecules which are relatively free to move between adjacent hydrogen-bonding sites, and a stabilization energy of approximately 8 kcal/mole per water molecule. Within the rigid dimer approximation, successive hydration of uracil is limited to the addition of one, two, or three water molecules.     

Discrimination of isomeric structures using information theoretic topological indices

Three newly defined information theoretic topological indices, namely “degree complexity (I^d),” “graph vertex complexity (H^V),” and “graph distance complexity (H^D)” along with three other information indices have been used to study their discriminating power of 45 trees and 19 monocyclic graphs. It is found that the newly defined indices have satisfactory discriminating power while H^D has been found to be the only index to discriminate all the graphs studied.     

Hydration of Sugars in the Gas Phase: Regioselectivity and Conformational Choice in N‐Acetyl Glucosamine and Glucose

The influence of an acetamido group in directing the preferred choice of hydration sites in glucosamine and a consequent extension of the working rules governing regioselective hydration and conformational choice, have been revealed through comparisons between the conformations and structures of “free” and multiply hydrated phenyl N‐acetyl‐β‐D ‐glucosamine (βpGlcNAc) and phenyl β‐D ‐glucopyranoside (βpGlc), isolated in the gas phase at low temperatures. The structures have been assigned through infrared ion depletion spectroscopy conducted in a supersonic jet expansion, coupled with computational methods. The acetamido motif provides a hydration focus that overwhelms the directing role of the hydroxymethyl group; in multiply hydrated βpGlcNAc the water molecules are all located around the acetamido motif, on the “axial” faces of the pyranose ring rather than around its edge, despite the equatorial disposition of all the hydrophilic groups in the ring. The striking and unprecedented role of the C‐2 acetamido group in controlling hydration structures may, in part, explain the differing and widespread roles of GlcNAc, and perhaps GalNAc, in nature.     

Method for construction of characteristic polynomials via graph linearization

A new method for construction of characteristic polynomials (CP) of complicated graphs having arbitrary edge and vertex weights has been developed. The method first converts the graph into isospectral linear chains with weighted vertices and edges and then builds up the CP coefficients recursively. Two types of graphs have been used for illustration, viz., (i) graphs that can be linearized by symmetry factorization and (ii) graphs without symmetry which are to be linearized by an algorithm involving walks of unit length. Both types have been illustrated, of which type (i) includes the Schlegel of fullerene fragment C₂₀ and another large graph with many fused rings. © 1997 John Wiley & Sons, Inc. Int J Quant Chem 65 : 199–204, 1997     

Polycations. III. Synthesis of polyphosphonium salts for use as antibacterial agents

Several series of polyphosphonium salts have been prepared. These include those in which the cationic sites are located at regular intervals along a linear chain of defined length (“strings”) and those in which the cationic sites are in linkages arrayed in branching arms about a central focus unit (“balloons”). Similarly, polyphosphonium balloon arrays have been attached to a polystyrene backbone providing a material that can serve as an anionic exchange resin. Several of the resultant materials have demonstrated antibacterial activity, in systems analogous to those previously reported for simple functionalized polystyrene species, and continue to be investigated. © 1998 John Wiley & Sons, Inc. Heteroatom Chem 9:495–502, 1998     

Molecular graphs as topological objects in space

Stereochemistry deals primarily with distinctions based on rigid geometry, e.g., bond angles and lengths. But some chemical species have molecular graphs (such as knots, catenanes, and nonplanar graphs K₅ and K_3.3) that reside in space in a topologically nontrivial way. For such molecules there is hope of using topological methods to gain chemical information. Viewing a molecular graph as a topological object in space makes it unrealistically flexible; but if one proves that a certain graph is “topologically chiral” or that two graphs are “topological diastereomers,” then one has ruled out interconversion under any physical conditions for which the molecular graph still makes sense. In this paper, we consider several kinds of topological questions one might ask about graphs in space, methology and results available, and specific topological properties of various molecules.