Core electrons and hydrogen atoms in chemical graph theory

General and complete graphs have recently been used to free chemical graph theory, and especially molecular connectivity theory, from spurious concepts, which belonged to quantum chemistry with no direct counterpart in graph theory. Both types of graph concepts allow the encoding of multiple bonds, non-bonding electrons, and core electrons. Furthermore, they allow the encoding of the bonded hydrogen atoms, which are normally suppressed in chemical graphs. This suppression could sometimes have nasty consequences, like the impossibility to differentiate between compounds, whose hydrogen-suppressed chemical graphs are completely equivalent, like for the CH₂F₂ and BHF₂ compounds. At the computational level the new graph concepts do not introduce any dramatic changes relatively to previous QSPR/QSAR studies. These concepts can nevertheless help in encoding the many electronic features of a molecule, achieving, as a bonus, an improved quality of the modeled properties, as it is here exemplified with a set of properties of different classes of compounds.     

Encoding the core electrons with graph concepts

The core electron problem of atoms in chemical graph studies has always been considered as a minor problem. Usually, chemical graphs had to encode just a small set of second row atoms, i.e., C, N, O, and F, thus, graph and, in some cases, pseudograph concepts were enough to "graph" encode the molecules at hand. Molecular connectivity theory, together with its side-branch the electrotopological state, introduced two "ad hoc" algorithms for the core electrons of higher-row atoms based, mainly, on quantum concepts alike. Recently, complete graphs, and, especially, odd complete graphs have been introduced to encode the core electrons of higher-row atoms. By the aid of these types of graphs a double-valued algorithm has been proposed for the valence delta, deltav, of any type of atoms of the periodic table with a principal quantum number n > or =2. The new algorithm is centered on an invariant suggested by the hand-shaking theorem, and the values it gives rise to parallel in some way the values derived by the aid of the two old "quantum" algorithms. A thorough comparative analysis of the newly proposed algorithms has been undertaken for atoms of the group 1A-7A of the periodic table. This comparative study includes the electronegativity, the size of the atoms, the first ionization energy, and the electron affinity. The given algorithm has also been tested with sequential complete graphs, while the even complete graphs give rise to conceptual difficulties. QSAR/QSPR studies do not show a clear-cut preference for any of the two values the algorithm gives rise to, even if recent results seem to prefer one of the two values.     

The hydrogen perturbation in molecular connectivity computations

A new algorithm for the delta(v) number, the basic parameter of molecular connectivity indices, is proposed. The new algorithm, which is centered on graph concepts like complete graphs and general graphs, encodes the information of the bonded hydrogen on different atoms through a perturbation parameter that makes use of no new graph concepts. The model quality of the new algorithm is tested with 13 properties of seven different classes of compounds, as well as with composite classes of compounds with the same property and with composite properties of the same class of compounds. Chosen properties and classes of compounds display different percentage of bonded hydrogen atoms, which allow a checking of the importance of this parameter. A comparison is drawn with previous results with zero contribution for the hydrogen perturbation as well as among results obtained by changing the number of compounds of a property but keeping constant the percentage of hydrogen atoms. Results underline the importance of the property as well as the importance of the number of compounds in determining the level of the hydrogen perturbation. Molecular connectivity terms are in some cases more critical than the combination of indices in detecting the perturbation introduced by the hydrogen atoms.     

Complete graph conjecture for inner-core electrons: homogeneous index case

The complete graph conjecture that encodes the inner-core electrons of atoms with principal quantum number n >or= 2 with complete graphs, and especially with odd complete graphs, is discussed. This conjecture is used to derive new values for the molecular connectivity and pseudoconnectivity basis indices of hydrogen-suppressed chemical pseudographs. For atoms with n = 2 the new values derived with this conjecture are coincident with the old ones. The modeling ability of the new homogeneous basis indices, and of the higher-order terms, is tested and compared with previous modeling studies, which are centered on basis indices that are either based on quantum concepts or partially based on this new conjecture for the inner-core electrons. Two similar algorithms have been proposed with this conjecture, and they parallel the two "quantum" algorithms put forward by molecular connectivity for atoms with n > 2. Nine properties of five classes of compounds have been tested: the molecular polarizabilities of a class of organic compounds, the dipole moment, molar refraction, boiling points, ionization energies, and parachor of a series of halomethanes, the lattice enthalpy of metal halides, the rates of hydrogen abstraction of chlorofluorocarbons, and the pED(50) of phenylalkylamines. The two tested algorithms based on the odd complete graph conjecture give rise to a highly interesting model of the nine properties, and three of them can even be modeled by the same set of basis indices. Interesting is the role of some basis indices all along the model.     

Fragment‐based similarity searching with infinite color space

Fragment‐based searching and abstract representation of molecular features through reduced graphs have separately been used for virtual screening. Here, we combine these two approaches and apply the algorithm RedFrag to virtual screens retrospectively and prospectively. It uses a new type of reduced graph that does not suffer from information loss during its construction and bypasses the necessity of feature definitions. Built upon chemical epitopes resulting from molecule fragmentation, the reduced graph embodies physico‐chemical and 2D‐structural properties of a molecule. Reduced graphs are compared with a continuous‐similarity‐distance‐driven maximal common subgraph algorithm, which calculates similarity at the fragmental and topological levels. The performance of the algorithm is evaluated by retrieval experiments utilizing precompiled validation sets. By predicting and experimentally testing ligands for endothiapepsin, a challenging model protease, the method is assessed in a prospective setting. Here, we identified five novel ligands with affinities as low as 2.08 μM. © 2015 Wiley Periodicals, Inc.     

Graph theoretical procedure for obtaining analytical expressions of eigenspectra of linear chains and cycles with alternant vertex weights and same edge weight: Application to some complicated graphs

A graph theoretical procedure for obtaining eigenvalues of linear chains and cycles having alternant vertex weights (h₁, h₂, h₁, h₂, h₁, h₂, …) and the same edge weight (k) have been developed. The eigenvalues of some complicated graphs, such as graphs of linear polyacenes, methylene‐substituted linear polyacenes and cylindrical polyacene strips, stack graphs, and reciprocal graphs have been shown to be generated in closed analytical forms by this procedure. Many such graphs represent chemically important molecules or radicals. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005     

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.     

Functional groups expressed as graphs extracted from the Laplacian of the electron density

The valence charge concentration shell, as determined by the Laplacian of the electron density, is used as a source of quantum topological graphs, called L‐graphs. A considerable number of such graphs are extracted from the ab initio wave functions of 31 molecules calculated at the B3LYP/6‐311+G(2d,p)//B3LYP/6‐311+G(2d,p) level, covering common functional groups in organic chemistry. We show how L‐graphs can be constructed from a largely transferable subgraph called atomic L‐graph. We investigate the topological stability of the L‐graphs as a function of the basis set. Reliable and consistent atomic L‐graphs are only obtained with basis sets of triple‐zeta quality or higher. The recurrence of invariant motifs or subgraphs in the L‐graphs enables the isolation of 16 atomic L‐graphs. © 2003 Wiley Periodicals, Inc. Int J Quantum Chem, 2003     

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.     

Conjugated circuits currents in hexabenzocoronene and its derivatives formed by joining proximal carbons

We report on calculated CC bond currents for a dozen derivatives of hexabenzocoroenene in which one or more proximal carbon atoms at the molecular periphery have been bridged. The approach that we use is graph‐theoretical in nature, following our outline of this method in 2003, which is based on finding all conjugated circuits in all Kekulé valence structures of these molecules. To the π‐electrons having 4n + 2 π‐electrons are assigned anticlockwise π‐electron currents and to conjugated circuits having 4n π‐electrons are assigned π‐electron currents. One may summarize the results reported in this work by stating that CC bond currents in the compounds considered decrease on going from peripheral rings to the central ring of the molecule, and also that CC bond currents decrease by insertion of bridges to proximal peripheral benzenoid rings. © 2012 Wiley Periodicals, Inc.     

Fibonacci relations. On the computation of some counting polynomials of very large graphs

A definition of a set of Fibonacci graphs is introduced which allows construction of several counting polynomials of very large graphs quite easily using a pencil-and-a-paper approach. These polynomials include matching, sextet, independence, Aihara and Hosoya polynomials. Certain combinatorial properties of Kekulé counts of benzenoid hydrocarbons are given. A relation to a new topological function that counts the cardinality of graph topology [23] is given.Dedicated to Professor Oskar E. Polansky for his enthusiastic support, participation and promotion of chemical graph theory.     

Molecular graph convolutions: moving beyond fingerprints

Molecular “fingerprints” encoding structural information are the workhorse of cheminformatics and machine learning in drug discovery applications. However, fingerprint representations necessarily emphasize particular aspects of the molecular structure while ignoring others, rather than allowing the model to make data-driven decisions. We describe molecular graph convolutions, a machine learning architecture for learning from undirected graphs, specifically small molecules. Graph convolutions use a simple encoding of the molecular graph—atoms, bonds, distances, etc.—which allows the model to take greater advantage of information in the graph structure. Although graph convolutions do not outperform all fingerprint-based methods, they (along with other graph-based methods) represent a new paradigm in ligand-based virtual screening with exciting opportunities for future improvement.     

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.