Generalized Hosoya polynomials of hexagonal chains

Similar to the well-known Wiener index, Eu et al. [Int. J. Quantum Chem. 106 (2006) 423–435] introduced three families of topological indices including Schultz index and modified Schultz index, called generalized Wiener indices, and gave the similar formulae of generalized Wiener indices of hexagonal chains. They also mentioned three families of graph polynomials in x, called generalized Hosoya polynomials in contrast to the (standard) Hosoya polynomial, such that their first derivatives at x = 1 are equal to generalized Wiener indices. In this note we gave explicit analytical expressions for generalized Hosoya polynomials of hexagonal chains.     

Structure and properties of organic molecules. 2. Graph-theoretical study of alkenes and alcohols

Topological indices for describing organic compounds with multiple bonds and heteroatoms are treated. The Schultz indices (including modified ones) and the connectivity indices of various types are calculated for saturated alkenes and alcohols. The discriminating ability of the indices and their applicability for structure-property correlations are examined. Various functions approximating the property-index correlations are tested using regression analysis. The Schultz and connectivity indices possess nearly the same correlation ability, as shown by reference to formation enthalpy, molar volume, evaporation heat, and boiling temperature. They may be effectively used for calculating and predicting the physicochemical properties of the title compounds. Tver State University. Translated fromZhurnal Strukturnoi Khimii, Vol. 39, No. 3, pp. 493–499, May–June, 1998. This work was supported by RFFR grant No. 96-03-32384.     

Structure and properties of organic molecules. 1. Three-dimensional topological indices of alkanes

The Winer and Schultz three-dimensional topological indices of alkanes are considered and compared with ordinary two-dimensional indices. Methods of index design and details of calculations are discussed. Unlike two-dimensional indices, three-dimensional ones adequately reflect differences between both structural and rotational isomers. Regression analysis of property-index correlations for formation enthalpy, molar volume, evaporation heat, and boiling temperature is performed using various two- and three-parameter (linear, quadratic, exponential, logarithmic, and power) functions. The ability of three-dimensional indices to correlate with these properties is found to be as good as that of two-dimensional ones. Tver State University. Translated fromZhurnal Strukturnoi Khimii, Vol. 39, No. 3, pp. 484–492, May–June, 1998. This work was supported by RFFR grant No. 96-03-32384.     

Generalized Wiener indices in hexagonal chains

The Wiener index, or the Wiener number, also known as the “sum of distances” of a connected graph, is one of the quantities associated with a molecular graph that correlates nicely to physical and chemical properties, and has been studied in depth. An index proposed by Schultz is shown to be related to the Wiener index for trees, and Ivan Gutman proposed a modification of the Schultz index with similar properties. We deduce a similar relationship between these three indices for catacondensed benzenoid hydrocarbons (graphs formed of concatenated hexagons, or hexagonal chains, or sometimes acenes). Indeed, we may define three families of generalized Wiener indices, which include the Schultz and Modified Schultz indices as special cases, such that similar explicit formulae for all generalized Wiener indices hold on hexagonal chains. We accomplish this by first giving a more refined proof of the formula for the standard Wiener index of a hexagonal chain, then extending it to the generalized Wiener indices via the notion of partial Wiener indices. Finally, we discuss possible extensions of the result. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2006     

Correlations of indices calculated from the matrix of maximal topological distances with boiling points of alkylbenzenes

The matrix of maximal topological distances of a graph can serve as a basis for constructing new topological indices of ringcontaining structures. A comparative study of the structureproperty correlations of the indices of the matrix of maximal topological distances and known Wiener, Horary, and Schultz indices (instructive sample of 29 alkylbenzenes) showed that the best two- and three-parameter correlations with boiling points include the indices of the maximal distance matrix. The two-parameter (r = 0.0992, s = 3.5) and three-parameter (r = 0.994, s = 3.1) correlations may be used for quantitative predictions of the boiling points of alkylbenzenes. Translated fromZhumal Struktumoi Khimii, Vol. 38, No. 1, pp. 167–172, January–February, 1997.     

Computing weighted Szeged and PI indices from quotient graphs

The weighted (edge-)Szeged index and the weighted (vertex-)PI index are modifications of the (edge-)Szeged index and the (vertex-)PI index, respectively, because they take into account also the vertex degrees. As the main result of this article, we prove that if G is a connected graph, then all these indices can be computed in terms of the corresponding indices of weighted quotient graphs with respect to a partition of the edge set that is coarser than the Θ^*-partition. If G is a benzenoid system or a phenylene, then it is possible to choose a partition of the edge set in such a way that the quotient graphs are trees. As a consequence, it is shown that for a benzenoid system, the mentioned indices can be computed in sublinear time with respect to the number of vertices. Moreover, closed formulas for linear phenylenes are also deduced.     

Application of the electronegativity indices of organic molecules to tasks of chemical informatics

An efficient structure filtration method for the operation with chemical databases containing information on the structures and properties of organic molecules was proposed. The technique involves the use of electronegativity indices for generation of identification keys and for isomorphism tests of the molecular graphs corresponding to the structural formulas. The test set for the method proposed included a total of 95,000,000 molecules containing up to sixty carbon atoms. Tests revealed a high discriminating capability of the electronegativity indices and high efficiency of the method for solving both general problems (recognition of chemical structures, chemical database management systems) and specific tasks (generation of molecular graphs, etc.) in chemical informatics. Dedicated to Academician N. S. Zefirov on the occasion of his 70th birthday. Published in Russian in Izvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 2166–2176, September, 2005.     

Graph and hypergraph models of molecular structure: A comparative analysis of indices

Eight series of molecular structures represented by molecular hypergraphs and molecular graphs are considered. A comparative analysis is performed for 23 integral topological and information theoretical indices for the graph and hypergraph models. For each series, the sensitivity is studied on both models, correlations are established for indices inside the models, and indices not correlating with each other are found. The results of this analysis showed that the sensitivity of most indices is higher in the hypergraph model. The total number of noncorrelated indices also increases in the latter model. Translated fromZhurnal Struktumoi Khimii, Vol. 39, No. 6, pp. 1163–1171, November–December, 1998.     

Graphs to chemical structures 3. General theorems with the use of different sets of sphericity indices for combinatorial enumeration of nonrigid stereoisomers

The extended sphericity indices of k-cycles, which were defined in Part 2 of this series (S. Fujita, Theor Chem Acc, Online: http://www.springerlink.com/index/10.1007/s00214-004-0606-z) according to the enantiospheric, homospheric, or hemispheric nature of each k-cycle, are further extended to prove more general theorems for enumerating nonrigid stereoisomers with rotatable ligands. One of the extended points is the use of different sets of sphericity indices to treat one or more orbits contained in skeletons and ligands. Another is to take account of chirality in proligands and sub-proligands, the latter of which are introduced to consider further inner structures of ligands. Two theorems for enumerating nonrigid stereoisomers are proved by adopting two schemes of their derivation, i.e., the scheme ``positions of a skeleton ⇐ proligands ⇐ ligands (positions of a ligand ⇐ sub-proligands)' and the scheme ``(positions of a skeleton ⇐ proligands ⇐ ligands (positions of a ligand)) ⇐ sub-proligands'. The theorems are applied to the stereoisomerism of trihydroxyglutaric acids. Thereby, it is demonstrated where Pólya's theorem and other previous methods are deficient, when applied to the enumeration of stereoisomers.     

Minimum general sum-connectivity index of unicyclic graphs

The general sum-connectivity index of a graph G is defined as χ _α (G) = ∑_edges (d _u + d _v ) ^α , where d _u denotes the degree of vertex u in G and α is a real number. In this report, we determine the minimum and the second minimum values of the general sum-connectivity indices of n-vertex unicyclic graphs for non-zero α ≥ −1, and characterize the corresponding extremal graphs.     

Graphs to chemical structures 5. Combinatorial enumeration of centroidal and bicentroidal three-dimensional trees as stereochemical models of alkanes

Three-dimensional trees (3D-trees), which are defined as a 3D version of trees, are enumerated by Fujita’s proligand method formulated in Part 1 to Part 3 of this series (Fujita in Theor Chem Acc 113:73–79, 113:80–86, 2005; 115:37–53, 2006). Such 3D-trees are classified into centroidal and bicentroidal 3D-trees, which correspond to respective promolecules having proligands as substituents. In order to enumerate such centroidal and bicentroidal 3D-trees, cycle indices with chirality fittingness (CI-CFs) are formulated as being composed of three kinds of sphericity indices, i.e., a _d for homospheric cycles, c _d for enantiospheric cycles, and b _d for hemispheric cycles. The CI–CFs are capable of giving itemized results with respect to chiral and achiral 3D-trees so that they are applied to derive functional equations (a(x), c(x ²), and b(x)). The generating functions of planted 3D-trees, which are formulated and calculated elsewhere, are introduced into such functional equations. Thereby, the numbers of 3D-trees or equivalently those of alkanes as stereoisomers are calculated and collected up to a carbon content of 20 in a tabular form. Now, the enumeration problem initiated by a mathematician Cayley (Philos Mag 47(4):444–446, 1874) has been solved in such a systematic and integrated manner as satisfying both mathematical and chemical requirements.     

Molecular modeling of wine polyphenols

Due to wide range of health effects of wine polyphenols, it is important to investigate the relationship between their structure and physical properties (quantitative structure–property relationship, QSPR). We have investigated linear, nonlinear (polynomial), and multiple linear relationships between given topological indices and molecular properties of main pharmacological active components of wine, such as molecular weight (MW), van der Waals volume (Vw), molar refractivity (MR), polar molecular surface area (PSA) and lipophilicity (log P). Partition coefficient (log P) was calculated using three different computer program (CLOGP, ALOGPS and MLOGP). The best models were achieved using the MLOGP program. Topological indices used for correlation analysis include: the Wiener index, W(G); connectivity indices, χ(G); the Balaban index, J(G); information-theoretic index, I(G); and the Schultz index, MTI(G). QSPR was performed on the set of 19 polyphenols and, particularly, on the group of phenolic acids, and on the group of flavonoids with resveratrol. The connectivity index has been successfully used for describing almost all parameters. Significant correlations were achieved between the Wiener index and van der Waals volume, as well as molecular weight.     

Unicycle Graphs with Maximum Generalized Topological Indices

Let G be an unicycle graph and d _v the degree of the vertex v. In this paper, we investigate the following topological indices for an unicycle graph , , where m ≥ 2 is an integer. All unicycle graphs with the largest values of the three topological indices are characterized. This research is supported by the National Natural Science Foundation of China(10471037)and the Education Committee of Hunan Province(02C210)(04B047).     

The calculation of electron localization and delocalization indices at the Hartree–Fock, density functional and post-Hartree–Fock levels of theory

 Localization, λ(A), and delocalization indices, δ(A,B), as defined in the atoms in molecules theory, are a convenient tool for the analysis of molecular electronic structure from an electron-pair perspective. These indices can be calculated at any level of theory, provided that first- and second-order electron densities are available. In particular, calculations at the Hartree–Fock (HF) and configuration interaction (CI) levels have been previously reported for many molecules. However, λ(A) and δ(A,B) cannot be calculated exactly in the framework of Kohn–Sham (KS) density functional theory (DFT), where the electron-pair density is not defined. As a practical workaround, one can derive a HF-like electron-pair density from the KS orbitals and calculate approximate localization and delocalization indices at the DFT level. Recently, several calculations using this approach have been reported. Here we present HF, CI and approximate DFT calculations of λ(A) and δ(A,B) values for a number of molecules. Furthermore, we also perform approximate CI calculations using the HF formalism to obtain the electron-pair density. In general, the approximate DFT and CI results are closer to the HF results than to the CI ones. Indeed, the approximate calculations take into account Coulomb electron correlation effects on the first-order electron density but not on the electron-pair density. In summary, approximate DFT and CI localization and delocalization indices are easy to calculate and can be useful in the analysis of molecular electronic structure; however, one should take into account that this approximation increases systematically the delocalization between covalently bonded atoms, with respect to the exact CI results. Received: 13 February 2002 / Accepted: 24 April 2002 / Published online: 18 June 2002     

On Unicycle Graphs with Maximum and Minimum Zeroth-order Genenal Randić Index

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 R _α⁰(G) = ∑ _v∈V(G) d _v ^α where α is an arbitrary real number. In this paper, we obtained the lower and upper bounds for the zeroth-order general Randić index R _α⁰(G) among all unicycle graphs G of order n. We give a clear picture for R _α⁰(G) of unicycle graphs according to real number α in different intervals.     

Analysis of the topological dependency of the characteristic polynomial in its chebyshev expansion

The structural dependency (effect of branching and cyclisation) of an alternative form, the Chebyshev expansion, for the characteristic polynomial were investigated systematically. Closed forms of the Chebyshev expansion for an arbitrary star graph and a bicentric tree graph were obtained in terms of the “structure factor” expressed as the linear combination of the “step-down operator”. Several theorems were also derived for non-tree graphs. Usefulness and effectiveness of the Chebyshev expansion are illustrated with a number of examples. Relation with the topological index (Z _G ) was discussed. Operated for the U.S. Department of Energy by ISU under contract no. W-ENG-7405-82. Supported in part by the Office of Director     

Graphs to chemical structures. 4: Combinatorial enumeration of planted three-dimensional trees as stereochemical models of monosubstituted alkanes

Planted three-dimensional (3D) trees, which are defined as a 3D version of planted trees, are enumerated by means of Fujita’s proligand method formulated in Parts 1–3 of this series [Fujita in Theor Chem Acc 113:73–79, 80–86, 2005; Fujita in Theor Chem Acc 115:37–53, 2006]. By starting from the concepts of proligand and promolecule introduced previously [Fujita in Tetrahedron 47:31–46, 1991], a planted promolecule is defined as a 3D object in which the substitution positions of a given 3D skeleton are occupied by a root and proligands. Then, such planted promolecules are introduced as models of planted 3D-trees. Because each of the proligands in a given planted promolecule is regarded as another intermediate planted promolecule in a nested fashion, the given planted promolecule is recursively constructed by a set of such intermediates planted promolecules. The recursive nature of such intermediate planted promolecules is used to derive generating functions for enumerating planted promolecules or planted 3D-trees. The generating functions are based on cycle indices with chirality fittingness (CI-CFs), which are composed of three kinds of sphericity indices (SIs), i.e., a _d for homospheric cycles, c _d for enantiospheric cycles, and b _d for hemispheric cycles. For the purpose of evaluating c _d recursively, the concept of diploid is proposed, where the nested nature of c _d is demonstrated clearly. The SIs are applied to derive functional equations for recursive calculations, i.e., a(x), c(x ²), and b(x). Thereby, planted 3D-trees or equivalently monosubstituted alkanes as stereoisomers are enumerated recursively by counting planted promolecules. The resulting values are collected up to 20 carbon content in a tabular form. Now, the enumeration problem initiated by mathematician Cayley [Philos Mag 47(4):444–446, 1874] has been solved in such a systematic and integrated manner as satisfying both mathematical and chemical requirements.     

The connective eccentricity index of graphs and its applications to octane isomers and benzenoid hydrocarbons

The connective eccentricity index (CEI) of a graph G is defined as , where ε_G(.) denotes the eccentricity of the corresponding vertex. The CEI obligates an influential ability, which is due to its estimating pharmaceutical properties. In this paper, we first characterize the extremal graphs with respect to the CEI among k-connected graphs (k-connected bipartite graphs) with a given diameter. Then, the sharp upper bound on the CEI of graphs with given connectivity and minimum degree (independence number) is determined. Finally, we calculate the CEI of two sets of molecular graphs: octane isomers and benzenoid hydrocarbons. We compare their CEI with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the CEI is better than or as good as the models corresponding to the other distance-based indices.     

On reciprocal molecular topological index

We report some properties of the reciprocal molecular topological index RMTI of a connected graph, and, in particular, its relationship with the first Zagreb index M₁. We also derive the upper bounds for RMTI in terms of the number of vertices and the number of edges for various classes of graphs, including K _r+1 -free graphs with r ≥ 2, quadrangle-free graphs, and cacti. Additionally, we consider a Nordhaus-Gaddum-type result for RMTI.