Classification of topologically chiral molecules

Graphs are partitioned into six classes from the perspective of chirality, depending on whether they are topologically achiral, whether there is at least one topologically achiral embedding, whether there is at least one rigidly achiral embedding, and whether there is at least one rigidly achiral presentation. Three of these classes are well represented by a variety of chemical structures: topologically chiral molecular graphs with no topologically achiral embeddings, topologically chiral molecular graphs with at least one rigidly achiral embedding, and topologically achiral molecular graphs with at least one rigidly achiral presentation. Known representatives of these three classes are described. Various meanings associated with the concepts molecular graph and intrinsic chirality are critically discussed. Previous arrangements of molecular graphs and molecules in a hierarchical order, ranging from the most to the least chiral, are interpreted in terms of the graph's and molecule's chiral persistence.     

Structural covariance of graphs

Algebraic structures including multiple rank tensors, linear and non-linear operators are related to and represented with various types of graphs. Special emphasis is placed on linear operators e.g. on the Hibert space. A different graph represents the same operator depending on the basis frame used, in general non-orthonormal. All such graphs are shown to belong in one equivalence class and are termed structurally covariant. Crucial indices related to eigenvalues but invariant under any basis frame changes including non-orthonormal ones provide one way to characterize each class. A set of rules are given that allow one to find the graphs structurally covarinat with a given one and/or to deduce the class indices directly by simple pictorial manipulations on a graph. Applications in various fields including the quantum theory of molecules and reactions are indicated.     

On the Ordering of a Class of Graphs with Respect to their Matching Numbers

Based on the number of k-matching m(G,k) of a graph G, Gutman and Zhang introduced an order relation of graphs: for graphs G ₁ and G ₂, if m(G ₁,k) ≥ m(G ₂,k) for all k. The order relation has important applications in comparing Hosaya indices and energies of molecular graphs and has been widely studied. Especially, Gutman and Zhang gave complete orders of six classes of graphs with respect to the relation . In this paper, we consider a class of graphs with special structure, which is a generalization of a class of graphs studied by Gutman and Zhang. Some order relations in the class of graphs are given, and the maximum and minimum elements with respect to the order relation are determined. The new results can be applied to order some classes of graphs by their matching numbers.     

Deformational covariance of graphs

Graphs on n vertices are classified into equivalence classes under the linear group L(n). All graphs representing the same operator on a vector space V _n belong to the same class. Graphs of different operators may or may not belong to the same class as ascertained by rules that were given by the author. Graphs in the same class are structurally covariant. If, in addition, two graphs can be continuously deformed into each other in the sense of varying line strengths while remaining structurally covariant throughout, then the two graphs are termed deformationally covariant along such paths. Applications in the quantum theory of chemistry and to the dynamic stability theory of coupled reaction systems which occur in various fields are indicated.     

Unicyclic Hückel molecular graphs with minimal energy

The minimal energy of unicyclic Hückel molecular graphs with Kekulé structures, i.e., unicyclic graphs with perfect matchings, of which all vertices have degrees less than four in graph theory, is investigated. The set of these graphs is denoted by such that for any graph in , n is the number of vertices of the graph and l the number of vertices of the cycle contained in the graph. For a given n(n ≥ 6), the graphs with minimal energy of have been discussed. MSC 2000: 05C17, 05C35     

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     

The adjoints of DNA graphs

In order to read a DNA sequence, we propose a method which induces the concept of DNA graph. In this paper, by discussing the adjoints of DNA graphs, we obtain more DNA graphs from known DNA graphs.AMS Subject classification: 92D20The Project Supported by Zhejiang Provincial Natural Science Foundation of China (102055) and The Foundation of Zhejiang Universities Youth Teachers.     

Isospectral benzenoid graphs with an odd number of vertices

A procedure for construction of isospectral pairs of benzenoid graphs is described. It is based on the Heilbronner wrapping procedure for construction of isospectral bipartite graphs. Only isospectral pairs having an odd number of vertices could be produced (the smallest among them has 33 vertices and 9 hexagons). Thus, the conjecture announced by Cioslowski is partially disproved.     

A Complete Solution to a Conjecture on the β-Polynomials of Graphs

In 1990, Gutman and Mizoguchi conjectured that all roots of the -polynomial (G,C,x) of a graph G are real. Since then, there has been some literature intending to solve this conjecture. However, in all existing literature, only classes of graphs were found to show that the conjecture is true; for example, monocyclic graphs, bicyclic graphs, graphs such that no two circuits share a common edge, graphs without 3-matchings, etc, supporting the conjecture in some sense. Yet, no complete solution has been given. In this paper, we show that the conjecture is true for all graphs, and therefore completely solve this conjecture.     

Intrinsic chirality of triple‐layered naphthalenophane and related graphs

We prove that the graph of triplelayered naphthalenophane and an infinite class of related graphs are all intrinsically chiral. We also give examples to illustrate that not all graphs which are contractible to a Möbius ladder with three rungs are necessarily intrinsically chiral.     

Dyad algebra and multiplication of graphs. I. Directed graphs

The ket-bra algebra for quantum mechanics and for the quantum chemistry.in valence shells was made by this author fully covariant recently. The resulting principle of linear covariance allowed diverse approaches such as molecular orbital, valence bond, localized orbital theories to come out as special cases in particular basis frames not necessarily orthonormal. The principal also led to the pictorial VIF (valency interaction formula) methods for deducing qualitative quantum chemistry directly from the structural formulas of molecules. The present set of two papers (II on undirected graphs) develops graphs and graph rules for abstract linear vector spaces, bras, kets, and abstract operators as ket-bra dyads. Multiplications of such operators are carried out with graphs of two kinds of lines and two kinds of vertices. The theorems are demonstrated on some examples and are useful, e.g., with the recent method of moments and in deriving Lie algebras pertinent to quantum chemistry.     

On zeroth-order general Randić index of conjugated unicyclic graphs

Let G be a graph and d _v denote the degree of the vertex v in G. The zeroth-order general Randić index of a graph is defined as where α is an arbitrary real number. In this paper, we investigate the zeroth-order general Randić index of conjugated unicyclic graphs G (i.e., unicyclic graphs with a perfect matching) and sharp lower and upper bounds are obtained for depending on α in different intervals.     

Experimental Evaluation of ΔS°rxn, ΔH°rxn, and ΔG°rxn Using Voltaic Cells: A First-Year College Chemistry Laboratory

Six voltaic cells have been evaluated for their suitability in the determination of thermodynamic parameters. The cells were prepared with all species present at conditions that approximate standard-state conditions and cell potentials were measured as a function of temperature. From these measurements graphs of voltage versus temperature were prepared. From these graphs it was possible to determine the standard heats of reaction (II_rxn), standard change in entropy or disorder (S°_rxn), and the Gibbs free energy (G°_rxn) for the spontaneous oxidation-reduction reactions. The standard cell potential (E°_cell values were also calculated. Two cells with opposite temperature dependence of the cell potential were found to produce good agreement with the literature values of G°_rxn, H°_rxn, S°^rxn and E°_cell. Criteria for identifying additional cells that may be suitable for potentiometric studies of thermodynamic parameters are also included.     

Unicyclic Graphs Possessing Kekulé Structures with Minimal Energy

Unicyclic graphs possessing Kekulé structures with minimal energy are considered. Let n and l be the numbers of vertices of graph and cycle C _l contained in the graph, respectively; r and j positive integers. It is mathematically verified that for and l = 2r + 1 or has the minimal energy in the graphs exclusive of , where is a graph obtained by attaching one pendant edge to each of any two adjacent vertices of C ₄ and then by attaching n/2 − 3 paths of length 2 to one of the two vertices; is a graph obtained by attaching one pendant edge and n/2 − 2 paths of length 2 to one vertex of C ₃. In addition, we claim that for has the minimal energy among all the graphs considered while for has the minimal energy.      

Semiautomatic determination of furazolidone and furaltadone by continuous and stopped flow FIA methods

New methods are proposed for the semiautomatic determination of furazolidone and furaltadone by photometric flow injection or stopped flow FIA analysis, on the basis of their alkaline hydrolysis reactions. Linear calibration graphs between 1 and 30 g/ml are obtained for both compounds, the detection limits for furazolidone and furaltadone being 0.20 and 0.26 g/ml, respectively, by flow injection and 0.27 and 0.21 g/ml by the stopped flow approach.     

Eigenvalues of large even annulenes in terms ofK 3,S 4 andS 5

Squares of the adjacency matrices of bipartite cycles (C_v) can be block-factored into matrices which correspond to vertex-weighted complete graphs forv = 6, vertex-weighted strongly regular graphs forv = 8 and 10, and vertex-weighted metrically regular graphs forv > 10. Using this fact and some properties of strongly and metrically regular graphs, it is shown that eigenvalues of large bipartite C _v graphs (i.e. large even annulenes) can be expressed by the general formula ± (2 ± (2 ± (... ± (2 +r _p)) ...), wherev = 2 ⁿ ×p,n is the number of surd () signs required andp = 3, 4 and 5. Here,r ₃,r ₄, andr ₅, are the eigenvalues of the complete graphK ₃ and the strongly regular graphsS ₄ andS ₅ respectively. The procedure does not require construction of characteristic polynomials for the determination of eigenvalues, and brings out a common topological origin for the two-fold degeneracies observed in the eigenvalue spectra of all even cycles and many odd cycles.     

Looking for Linearity: Integrating Graphing for First-Year Chemistry Students

Based on our observation that the general literature does not provide an organizing principle for the graphs that science students encounter, an approach called Looking for Linearity has been described. This approach is based on the hypothesis that when scientists look at their data and begin to represent it, they initially look for linearity. This is to say that scientists use Occams Razor; variables are used and transformed in such ways that when plotted against each other, the simplest representation—the straight line—is produced. A brief review of the topics typically covered in the first year of chemistry reveal a substantial number of relationships either expressed in the form of a straight line (gas laws, free energy, rate laws) or in terms of ratios that when graphed produce straight lines (density, specific heat capacity, stoichiometry). Looking for Linearity is an approach to graphing that serves four purposes for teaching first year chemistry students: 1) it weaves a common theme or thread through the entire year of General Chemistry, 2) it allows students to work like scientists, 3) it connects an important mathematical construct with chemical concepts, and 4) it provides a method to process data in other scientific fields like physics. The linearity heuristic is represented in what is called a Graphing Decision Tree. This tree shows, in simplified terms, how linearity can be used to organize different types of graphs found in the first year of chemistry. The Decision Tree is hierarchically structured from simple to increasing graphing complexity. Straight lines were listed as being the simplest to interpret, followed by exponential curves and then non-exponential curves; exponential curves were second because they could be converted to straight lines by using logarithms. Each pathway ends with examples of some of the different types of graphs our students will encounter in the first year of chemistry.     

Reaction graphs and a construction of reaction networks

Summary A general definition of reaction graphs is presented. For a pair of isomeric molecular graphs and , related by a chemical transformation , the reaction graph is determined using a maximal common subgraph defined for vertex mapping . A binary operation defined for graphs constructed over the same vertex set enables us to decompose the reaction graph into the sum of prototype reaction graphs. A decomposition of an overall reaction graph can be advantageously used for the construction of a reaction network. An oriented path in this network beginning at and ending at corresponds to a breakdown of the transformation into a sequence of intermediates.     

Spectrophotometric study of reactions of substituted phenols with MBTH in alkaline medium: The effect of phenol structure on the formation of analytically useful coloured products

The oxidative coupling reactions of unsubstituted phenol, 4-cresol, 2,6-xylenol and another fourteen substituted phenols with MBTH in alkaline medium containing potassium hexacyanoferrate (III) as the oxidant were examined by spectrophotometry. The maximum yield and stability of coloured species was attained at optimum pH 9.0±0.5. Stoichiometry of the reactions involving a four-electron transfer was confirmed. The coloured products were stable in an aqueous alkaline medium containing 50% (V/V) of a water-miscible organic solvent and they could be quantitatively extracted into chloroform. Absorption maxima _max of the reaction products ranged between 495 nm and 610 nm depending on the phenol structure and their molar absorptivities fell within the range 2,500–47,000L mol^–1 cm^–1 in aq. 50% ethanol. The calibration graphs were rectilinear for 2 to 20 M phenol or 2,6-xylenol (r = 0.9994–0.9997; n = 6) and the RSD values were 1.8% (n=10) when determining 10 mol L^–1 of the analytes. The reactivity of the phenols with MBTH (and hence the yield of the coloured species in alkaline medium) depending on the analyte structure decreased in the order: 2,6-dialkylphenols or 4-halogenated 2,6-dialkylphenols>2-alkyl, 2-alkoxy or 2-arylphenols and l-naphthol>unsubstituted phenol or 2-naphthol2-halogenated phenols>4-alkylphenols and 4-halogenated phenols>2-nitrophenol>2,4-or 3,4-dialkylphenols. For some 4-halogenated phenols the elimination of halogen upon coupling with the MBTH was observed.     

Quantum chemical reaction networks,reaction graphs and the structure of potential energy hypersurfaces

Global properties of the Born-Oppenheimer energy expectation value functional, defined over the nuclear configuration space R, are analyzed. Quantum chemical reaction graphs and reaction networks are defined in terms of intersection graphs of connected sets of nuclear geometries, representing various chemical structures. The set of all possible reaction mechanisms on the given energy hypersurface and the associated activation energy conditions are analyzed using reachability matrices defined over digraphs D ^s() and D ^s(, E).