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.     

Theorems for carbon cages

New theorems are established for cages (or polyhedra) with trivalent vertices. One theorem says that all such cages have at least three Kekulé structures (or perfect matchings). Thence, resonance generally appears as a possibility. Another theorem says that for every even vertex count >70 there is at least one cage of a preferable subclass, while for vertex count <70 the sole preferable cage is that of the truncated icosahedron. Thence, the unique role of the buckminsterfullerene structure for C₆₀ is mathematically indicated.[/p]Work supported by the Welch Foundation of Houston, Texas.     

Details of the reaction graphs for intramolecular isomerizations of the carboranes

From proposed mechanisms for framework reorganizations of the carboranes C₂B _n-2H _n ,n = 5–12, we present reaction graphs in which points or vertices represent individual carborane isomers, while edges or arcs correspond to the various intramolecular rearrangement processes that carry the pair of carbon heteroatoms to different positions within the same polyhedral form. Because they contain both loops and multiple edges, these graphs are actually pseudographs. Loops and multiple edges have chemical significance in several cases. Enantiomeric pairs occur among carborane isomers and among the transition state structures on pathways linking the isomers. For a carborane polyhedral structure withn vertices, each graph hasn(n -1)/2 graph edges. The degree of each graph vertex and the sum of degrees of all graph vertices are independent of the details of the isomerization mechanism. The degree of each vertex is equal to twice the number of rotationally equivalent forms of the corresponding isomer. The total of all vertex degrees is just twice the number of edges orn(n - 1). The degree of each graph vertex is related to the symmetry point group of the structure of the corresponding isomer. Enantiomeric isomer pairs are usually connected in the graph by a single edge and never by more than two edges.     

The graph center concept for polycyclic graphs

While the concept of the graph center is unambiguous (and quite old) in the case of acyclic graphs, an attempt has been made recently to extend the concept to polycyclic structures using the distance matrix of a graph as the basis. In this work we continue exploring such generalizations considering in addition to the distance matrix, self-avoiding walks or paths as graph invariants of potential interest for discriminating distinctive vertex environments in a graph of polycyclic structures. A hierachy of criteria is suggested that offers a systematic approach to the vertex discrimination and eventually establishes in most cases the graph center as a single vertex, a single bond (edge), or a single group of equivalent vertices. Some applications and the significance of the concept of the graph center are presented.     

Geometry and chemical bonding in polyhedral boranes, metallaboranes, and dimetallaboranes: From closo to isocloso to oblatocloso polyhedra

The geometry and chemical bonding in the closo metal-free boranes and the isoelectronic carboranes and C₂B_n−2H_n with 2n + 2 skeletal electrons are based on the most spherical deltahedra with a preference for degree 5 vertices, particularly for the boron atoms. Such deltahedral boranes can be considered to be three-dimensional aromatic systems, as indicated by strongly diatropic nucleus independent chemical shift values for (n = 6, 8, 9, 12). Metallaborane structures, particularly those with 9-11 vertices and only 2n rather than 2n + 2 apparent skeletal electrons, are often based on isocloso deltahedra with the metal atom at a degree 6 vertex. Dimetallaborane structures, particularly the rhenium derivatives Cp₂Re₂B_n−2H_n−2 (8 ? n ? 12), are based on highly non-spherical and very oblate deltahedra with the metal atoms typically at degree 6 or 7 vertices, which are the lowest curvature sites of the deltahedra. A viable model for the skeletal bonding in such dimetallaboranes can be developed if each of the two metal vertices is assumed to contribute five internal orbitals to the skeletal bonding. This leads to 2n + 4 skeletal electrons, which are partitioned into n surface bonds and a formal metal-metal double bond inside the oblate deltahedron.     

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.     

A new approach for devising local graph invariants: Derived topological indices with low degeneracy and good correlation ability

A new approach is presented for obtaining graph invariants which have very high discriminating ability for different vertices within a graph. These invariants are obtained as the solution set (local invariant set, LOIS) of a system of linear equationsQ · X = R, whereQ is a topological matrix derived from the adjacency matrix of the graph, andR is a column vector which encodes either a topological property (vertex degree, number of vertices in the graph, distance sum) or a chemical property (atomic number). Twenty examples of LOOIs are given and their degeneracy and ordering ability of vertices is discussed. Interestingly, in some cases the ordering of vertices obtained by means of these invariants parallels closely the ordering from an entirely different procedure based on Hierarchically Ordered Extended Connectivities which was recently reported. New topological indices are easily constructed from LOISs. Excellent correlations are obtained for the boiling points and vaporization enthalpies of alkanesversus the topological index representing the sum of local vertex invariants. Les spectacular correlations with NMR chemical shifts, liquid phase density, partial molal volumes, motor octane numbers of alkanes or cavity surface areas of alcohols emphasize, however, the potential of this approach, which remains to be developed in the near future.     

Defective vertices in arachno borane networks

Triangulated boron networks can be described in terms of the deviation of their local vertex environments from the degree 5 vertices found in ideal icosahedra. Vertices of degrees other than 5 or equivalent are considered to be defective vertices. This method, which was previously applied to deltahedral borane anions B(n)H(n)(2-) and nido-B(n)H(n+4) boranes, has now been applied to arachno boranes of the types B(n)H(n+6) and B(n)H(n+5)(-) (4 < or = n < or = 10). The known structures of the neutral arachno boranes B(4)H(10), B(8)H(14), and n-B(9)H(15) consist of triangulated boron networks with no defective vertices in accord with their higher stabilities relative to other neutral arachno boranes. In other structures of known arachno boranes, there are relatively small numbers of defective vertices, and these are isolated as far as possible from each other.     

Improved strategy in analytic surface calculation for molecular systems: Handling of singularities and computational efficiency

Computer methods for analytic surface calculations of molecular systems suffer from numerical instabilities and are CPU time consuming. In this article, we present proposals toward the solution of both problems. Singularities arise when nearly collinear triples of neighboring atoms or multiple vertices are encountered during the calculation. Topological decisions in analytic surface calculation algorithms (accessibility of vertices and arcs) are based upon the comparison of distances or angles. If two such numbers are nearly equal, then currently used computer programs may not resolve this ambiguity correctly and can subsequently fail. In this article, modifications in the analytic surface calculation algorithm are described that recognize singularities automatically and treat them appropriately without restarting parts of the computation. The computing time required to execute these alterations is minimal. The basic modification consists in defining an accuracy limit within which two values may be assumed as equal. The search algorithm has been reformulated to reduce the computational effort. A new set of formulas makes it possible to avoid mostly the extraction of square roots. Tests for small-and medium-sized intersection circles and for pairs of vertices with small vertex height help recognize fully buried circles and vertex pairs at an early stage. The new program can compute the complete topology of the surface and accessible surface area of the protein crambin in 1.50–4.29 s (on a single R3000 processor of an SGI 4D/480) depending on the compactness of the conformation where the limits correspond to the fully extended or fully folded chain, respectively. The algorithm, implemented in a computer program, will be made available on request. © John Wiley & Sons, Inc.     

Recent Progress in the Chemistry of Supercarboranes

The chemistry of boron clusters has been dominated by icosahedral carboranes for over half a century. Only in recent years has significant progress been made in the chemistry of supercarboranes (carboranes with more than 12 vertices). A number of CAd (carbon‐atoms‐adjacent) 13‐ and 14‐vertex carboranes, and CAp (carbon‐atoms‐apart) 13‐vertex carboranes as well as their corresponding 14‐ and 15‐vertex metallacarboranes have been successfully prepared and structurally characterized. This breakthrough relied on the use of CAd nido‐carborane dianions as starting materials. These supercarboranes can undergo single‐electron reduction to give stable supercarborane radical monoanions with [2n+3] framework electrons, and electrophilic substitution reaction to afford hexasubstituted supercarboranes. They can react with nucleophiles to offer monocarba‐closo‐dodecaborate monoanions from cage‐carbon extrusion reactions. Their unique chemical properties make the chemistry of supercarboranes distinct from that of their 12‐vertex analogues. These studies open up new possibilities for the development of polyhedral clusters of extraordinary size. This focus review offers an overview of recent advances in this growing research field.     

Reticular Synthesis of One-Dimensional Covalent Organic Frameworks with 4-c sql Topology for Enhanced Fluorescence Emission

Covalent organic frameworks (COFs) have been extensively investigated due to their unique structure, porosity, and functionality. However, at the topological level, COFs remain as two-dimensional (2D) or three-dimensional (3D) structures, while COFs with one-dimensional (1D) topology have not been systematically explored. In this work, we proposed a synthetic strategy for the construction of 1D-COFs based on non-linear edges and suitable high-symmetry vertices. Compared with their 2D-COFs counterparts, the 1D-COFs with AIEgens located at the vertex of the frame exhibited enhanced fluorescence. The density functional theory (DFT) calculations revealed that the dimensional-induced rotation restriction (DIRR) effect could spontaneously introduce additional non-covalent interactions between the strip frames, which could substantially diminish non-radiative transitions. This work also provides protocols for the design of 1D-COFs and a guidance scheme for the synthesis of emitting COFs.     

Limited occurrence of isocloso deltahedra with 9 to 12 vertices in low-energy hypoelectronic diferradicarbaborane structures

Theoretical studies show that the 10-vertex system Cp(2)Fe(2)C(2)B(6)H(8) is the only one of the 2n skeletal electron Cp(2)Fe(2)C(2)B(n-4)H(n-2) systems (n = 9, 10, 11, 12) for which a true isocloso deltahedron having a single degree 6 vertex is highly favored over alternative structures. This is demonstrated by the occurrence of only the 10-vertex isocloso deltahedron as the central Fe(2)C(2)B(6) polyhedron in all nine of the Cp(2)Fe(2)C(2)B(6)H(8) structures within 8 kcal/mol of the global minimum. Low energy isocloso structures are also observed for the 11-vertex Cp(2)Fe(2)C(2)B(7)H(9). However, interspersed with these isocloso structures are Cp(2)Fe(2)C(2)B(7)H(9) structures based on deltahedra having two or more degree 6 vertices. For the 12-vertex Cp(2)Fe(2)C(2)B(8)H(10), the six lowest energy structures all have central Fe(2)C(2)B(8) deltahedra with two degree 6 vertices, one for each iron atom. The Cp(2)Fe(2)C(2)B(8)H(10) structures having a central Fe(2)C(2)B(8) icosahedron with all degree 5 vertices lie at significantly higher energies, starting at 17.8 kcal/mol above the global minimum. The 9-vertex Cp(2)Fe(2)C(2)B(5)H(7) system appears to be too small for isocloso structures to be favorable, although three such structures are found at energies between 5.5 and 8.0 kcal/mol above the global minimum. Five Cp(2)Fe(2)C(2)B(5)H(7) structures based on the tricapped trigonal prism lie in an energy below the lowest energy isocloso structure. The lowest energy Cp(2)Fe(2)C(2)B(5)H(7) structure and two higher energy structures within 8.0 kcal/mol of the global minimum have central Fe(2)C(2)B(5) deltahedra with a degree 6 vertex for each iron atom.     

Exocyclobutadieneite

This communication describes a novel structural-type that could be the basis for a potentially new allotrope of C. The novel structural-type is called “exocyclobutadieneite”, and it is thus named for the 1,3-dimethylenecyclobutane generating fragment that the lattice is based upon. It is a 3-,4-connected network consisting of slightly distorted tetrahedral vertices, and slightly distorted pairs of trigonal planar vertices. The lattice can be derived from a known mineral structure called Cooperite (PtS or PdO) by a topologically isomorphic substitution of trigonal planar atom pairs, for square planar vertexes, in the parent Cooperite unit cell. As such, the new pattern bears a distinct counterpoint relationship with its sibling structural-type called the glitter lattice, which has already been described by the authors in several other papers. And whereas glitter is generated by a topologically isomorphic substitution of trigonal planar atom pairs for the square planar vertices in the Cooperite unit cell, in a fashion that extends the unit cell vertically along the crystallographic c-axis, the exocyclobutadieneite structure is generated, in counterpoint, by such an isomorphic substitution that extends the unit cell horizontally along the a- and b-axes. Both the resulting glitter and exocyclobutadieneite structural-types possess AB $_{2}$ stoichiometry (where A is a tetrahedral vertex and B is a trigonal planar vertex) and both occur in the tetragonal symmetry space group P4 $_{2}$ /mmc (#131). The networks differ, however, in the special positions adopted by the vertices in the resultant unit cells, and in their respective topology, as is evidenced by consideration of the Wells point symbols and Schläfli symbols for the 2 tetragonal networks. These differences are illustrated further in the course of the discussion to follow.     

Two metrics for a graph-theoretical model of organic chemistry

A graph-theoretical model of organic chemistry is proposed. The main idea behind this model is a molecular graph in the form of a multigraph with loops; its vertices are evaluated by vertex labels (atomic symbols). The chemical distance between two graphs from the same family of isomeric graphs is based on the maximal common subgraph. The produced reaction graph is composed of the minimal number of edges and/or loops. The reaction distance assigned to the chemical transformationG ₁ G ₂ is equal to the minimal number of the so-called elementary transformations that are necessary for the transformation ofG ₁ intoG ₂ Because these metrics are not isometric, the resulting reaction graphs may depend on the metric used.Dedicated to the memory of Professor Milan Sekanina     

CB11Me11 boronium ylides: carba-closo-dodecaboranes with a naked boron vertex

In pentane solution, 2 equiv of the icosahedral CB(11)Me(12)(*) radical cleaves the Si-Si bond of hexaalkyldisilanes by boron to silicon methyl transfer with formation of 2 equiv of methyltrialkylsilanes. The loss of a methyl radical converts the CB(11)Me(12)(*) radical into an internally charge-compensated "boronium ylide" CB(11)Me(11) with a naked vertex, which can be formally viewed as a deprotonated hypercloso carborane. It has been isolated as an air-sensitive solid, stable only below approximately -60 degrees C. The naked vertex appears to be in position 12 since the material reacts instantaneously with alcohols and ethers to form the 12-alkoxy anions 12-CB(11)Me(11)OR. It reacts with many other nucleophiles to yield more complex mixtures containing similar products. DFT calculations for the four CB(11)Me(11) isomers give closed-shell ground-state electronic structures. For the isomer with naked vertex 12, a DFT computational search failed to reveal any skeletal dimers, apparently due to excessive methyl-methyl repulsions, and only a cyclic dimer bound through weak interactions of one of the 7-methyl hydrogen atoms on each cage with the empty exocyclic orbitals on B12 of the other cage was found. Natural hybrid orbital populations suggest that the three possible isomers of monomeric boronium ylides are close to true singlet ylides, with triplet states approximately 50 kcal/mol higher. The calculated electronic structure of the carbonium ylide is close to a singlet carbene, with a triplet state approximately 16 kcal/mol higher. An attempted preparation of Me(3)C(+)CB(11)Me(12)(-) yielded neopentane and products consistent with a sequential loss of methyl groups from the carborane cage with a transient formation of similar boronium ylides. Probable mechanisms of these methyl transfer reactions are considered, and the possibly quite general role of "ylide" structures in Lewis acid induced substitution reactions on the boron vertices of carboranes and boranes is noted.     

Effect of trifluoroacetic acid on the reduction of disulfide bridges in peptides analyzed by fast-atom bombardment mass spectrometry

Some of the factors that influence the reduction of disulfide-containing peptides under fast-atom bombardment have been investigated using two neurohormonal peptides that include disulfide bridges in their structures. Deaminoarginine-vasopressin (DAVP) and arginine-vasopressin (AVP) have been analyzed as their acetate and trifluoroacetate salts. Results obtained in a thioglycerol matrix indicate that the peptides analyzed as their acetate salts are completely reduced under bombardment, whereas the trifluoroacetate salts show little evidence of reduction. Addition of trifluoroacetic acid to the acetate sample prior to bombardment inhibits reduction whereas addition after bombardment shows no effect on the reduction, thereby indicating the irreversibility of the process. Time-monitoring experiments conducted with the acetate salts of DAVP and AVP in common matrices such as thioglycerol, dithiothreitol + diethioerythritol, glycerol, hydroxyethyldisulfide and nitrobenzyl-alcohol demonstrate an important effect of the chemical nature of the matrix on reduction. In matrices containing thiol groups, the reduction is extensive, whereas it is almost suppressed in matrices such as hydroxyethyldisulfide and nitrobenzylalcohol. However, the addition of trifluoroacetic acid to all of these matrices essentially eliminates reduction and provides measured isotopic peak ratios that are in agreement with theoretically calculated values for these peptides.     

Chemical applications of topology and group theory

Earlier approaches to the analysis of chemical dynamic systems using kinetic logic are refined to deal more effectively with systems having the two or more feedback circuits required for chaos. The essential kinetic features of such a system can be represented by a directed graph (called aninfluence diagram) in which the vertices represent the internal species and the directed edges represent kinetic relationships between the internal species. Influence diagrams characteristic of chaotic chemical systems have the following additional features: (1) They are connected; (2) Each vertex has at least one edge directed towards it and one edge directed away from it; (3) There is at least one vertex, called a turbulent vertex, with at least two edges directed towards it. From such an influence diagram a state transition diagram representing the qualitative dynamics of the system can be obtained using the following 4-step procedure: (1) A logical relationship is assigned at each turbulent vertex; (2) A local truth table is generated for each circuit in the influence diagram; (3) The local truth tables are combined to give a global truth table using the logical relationships at the turbulent vertices; (4) The global truth table is used to determine the corresponding state transition diagram using previously described methods. This refined procedure leads to a more restricted set of influence diagrams having the interlocking cycle flow topology required for chaos than the procedure described earlier. Systems with 3 internal species are examined in detail using the refined procedure. All systems with 3 dynamic variables shown in the simulation studies of Rössler to give chaotic dynamics correspond to influence diagrams which give inter-locking cycle (chaotic) flow topologies by the refined procedure. In addition, two models for the Belousov-Zhabotinskii reaction are examined using the refined procedure. The results are potentially informative concerning possible mechanisms for the limitation of the accumulation of autocatalytically produced HBrO₂ (one of the internal species) during the course of this reaction.     

Graphs whose vertices are graphs with bounded degree: Distance problems

By an f-graph we mean a graph having no vertex of degree greater than f. Let U(n,f) denote the graph whose vertex set is the set of unlabeled f-graphs of order n and such that the vertex corresponding to the graph G is adjacent to the vertex corresponding to the graph H if and only if H is obtainable from G by either the insertion or the deletion of a single edge. The distance between two graphs G and H of order n is defined as the least number of insertions and deletions of edges in G needed to obtain H. This is also the distance between two vertices in U(n,f). For simplicity, we also refer to the vertices in U(n,f) as the graphs in U(n,f). The graphs in U(n,f) are naturally grouped and ordered in levels by their number of edges. The distance nf/2 from the empty graph to an f-graph having a maximum number of edges is called the height of U(n,f). For f =2 and for f≥(n-1)/2, the diameter of U(n,f) is equal to the height. However, there are values of the parameters where the diameter exceeds the height. We present what is known about the following two problems: (1) What is the diameter of U(n,f) when 3≥f<(n-1)/2? (2) For fixed f, what is the least value of n such that the diameter of U(n,f) exceeds the height of U(n,f)?