Wiener and Hyper-Wiener Numbers in a Single Matrix

Synthesis, photoinduced amination and topological indices of novel porphyrin dyads

Transition-metal-catalyzed homo and hetero coupling is a rapidly growing area of research. This work refers to the nickel-catalyzed photoinduced amination study of [Formula: see text]-bromotetraarylporphyrin and its Ni(II), Zn(II) and Cu(II) complexes. The selective mono [Formula: see text]-bromination of 5,10,15,20-tetrakis(4[Formula: see text]-isopropylphenyl)porphyrin was achieved with [Formula: see text]-bromosuccinimide. Under similar conditions, [Formula: see text]-bromination of Ni(II), Zn(II) and Cu(II) complexes of 5,10,15,20-tetrakis(4[Formula: see text]-isopropylphenyl)porphyrin successfully afforded the corresponding 2-bromometalloporphyrins. The [Formula: see text]-bromoporphyrin/metalloporphyrins were coupled with three different amines through the creation of the C–N bond by using an economical and air-tolerant photoactive catalyst (NiBr[Formula: see text] · 3H2O) at room temperature under 365 nm radiations. Nickel-catalyzed amination yields are compared with the traditional Buchwald–Hartwig amination yields. Due to the low operational cost, photoinduced nickel-catalyzed C–N couplings were found to be more economical than the Buchwald–Hartwig amination procedure, although the latter afforded higher yields. The nickel-catalyzed photoamination reaction was also extended for the one-pot synthesis of pyridine-3,5-diamine bridged porphyrin dyad. The intramolecular cyclization of the pyridine-3,5-diamine-bridged porphyrin dyad afforded a novel quinolino-fused porphyrin dyad. Degree- and distance-based topological indices of the newly synthesized porphyrins were calculated and correlated with their molar refractivity. All newly synthesized porphyrins are characterized by UV-vis, FTIR, 1H NMR, elemental analysis and mass spectrometry.

Remarks on Distance Based Topological Indices for ℓ-Apex Trees

A graph G is called an ℓ-apex tree if there exist a vertex subset A ⊂ V ( G ) with cardinality ℓ such that G − A is a tree and there is no other subset of smaller cardinality with this property. In the paper, we investigate extremal values of several monotonic distance-based topological indices for this class of graphs, namely generalized Wiener index, and consequently for the Wiener index and the Harary index, and also for some newer indices as connective eccentricity index, generalized degree distance, and others. For the one extreme value we obtain that the extremal graph is a join of a tree and a clique. Regarding the other extreme value, which turns out to be a harder problem, we obtain results for ℓ = 1 and pose some open questions for higher ℓ. Symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including topological indices of graphs.

Mold(2), molecular descriptors from 2D structures for chemoinformatics and toxicoinformatics

Research applications in chemoinformatics and toxicoinformatics increasingly use representations of molecules in the form of numerical descriptors that capture the structural characteristics and properties of molecules. These representations are useful for ADME/toxicity prediction, diversity analysis, library design, QSAR/QSPR, virtual screening, and other purposes. Molecular descriptors have ranged from relatively simple forms calculated from simple two-dimensional (2D) chemical structures to more complex forms representing three-dimensional (3D) chemical structures or complex molecular fingerprints consisting of numerous bit positions to represent specific chemical information. The Mold (2) software was developed to enable the rapid calculation of a large and diverse set of descriptors encoding two-dimensional chemical structure information. Comparative analysis of Mold (2) descriptors with those calculated by Cerius (2), Dragon, and Molconn-Z on several data sets using Shannon entropy analysis demonstrated that Mold (2) descriptors convey a similar amount of information. In addition, using the same classification method, slightly better models were generated using Mold (2) descriptors compared to those generated using descriptors from the compared commercial software packages. The low computing cost for Mold (2) makes it suitable not only for small data sets, such as in QSAR, but also for large databases in virtual screening. High reproducibility and reliability are expected because Mold (2) does not require 3D structures. Mold (2) is freely available to the public ( http://www.fda.gov/nctr/science/centers/toxicoinformatics/index.htm).

Note of the hyper-Wiener index

The hyper-Wiener index WW of a chemical tree T is defined as the sum of the products n1n2, over all pairs ?,? of vertices of T, where n1 and n2 are the number of vertices of T, lying on the two sides of the path which connects ? and ?. We examine a slight modification WWW of the hyper-Wiener index defined as the sum of the products n1n2n3, over all pairs ?,? of vertices of T, where n3 is the number of vertices of T, lying between ? and ?. It is found that WWW correlates significantly better with various physico-chemical properties of alkanes than WW. Lower and upper bounds for WWW, and an approximate relation between WWW and WW are obtained.

Quantitative structure-retention relationships for gas chromatographic retention indices of alkylbenzenes with molecular graph descriptors

Quantitative structure-retention relationships (QSRR) represent statistical models that quantify the connection between the molecular structure and the chromatographic retention indices of organic compounds, allowing the prediction of retention indices of novel, not yet synthesized compounds, solely from their structural descriptors. Using multiple linear regression, QSRR models for the gas chromatographic Kováts retention indices of 129 alkylbenzenes are generated using molecular graph descriptors. The correlational ability of structural descriptors computed from 10 molecular matrices is investigated, showing that the novel reciprocal matrices give numerical indices with improved correlational ability. A QSRR equation with 5 graph descriptors gives the best calibration and prediction results, demonstrating the usefulness of the molecular graph descriptors in modeling chromatographic retention parameters. The sequential orthogonalization of descriptors suggests simpler QSRR models by eliminating redundant structural information.

Peptide property modeling by Cluj indices

The novel Cluj property indices are used for modeling the biological properties of dipeptides: the ACE inhibition activity of a set of 58 dipeptides and the bitter tasting activity of a set of 48 dipeptides, taken from the literature. The results are compared to those reported in some previous works.

Quantitative structure-property relationships generated with optimizable even/odd Wiener polynomial descriptors

Chemical structures of organic compounds are characterized numerically by a variety of structural descriptors, one of the earliest and most widely used being the Wiener index W, derived from the interatomic distances in a molecular graph. Extensive use of distance-based structural descriptors or topological indices has been made in QSPR and QSAR models, drug design, toxicology, virtual screening of combinatorial libraries, similarity and diversity assessment. Novel topological indices are introduced representing a partitioning of the Wiener polynomial based on counts of even and odd molecular graph distances. During the QSAR/QSPR modeling process the variables of the even and odd power functions are optimized in order to offer an improved mapping of the investigated property. These novel topological indices are tested in QSPR models for the boiling temperature, molar heat capacity, standard Gibbs energy of formation, vaporization enthalpy, refractive index, and density of alkanes. In many cases, the even/odd Wiener polynomial indices proposed here give notably improved correlations or suggest simpler QSPR models.

QSAR comparative study of Wiener descriptors for weighted molecular graphs

Quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies use statistical models to compute physical, chemical, or biological properties of a chemical substance from its molecular structure, encoded in a numerical form with the aid of various descriptors. Structural indices derived from molecular graph matrices represent an important group of descriptors used in QSPR and QSAR models; recently, their utilization was extended to molecular similarity and diversity, in database mining and virtual screening of combinatorial libraries. Initially defined from the distance matrix, the Wiener index W was the source of novel graph descriptors derived from recently proposed molecular matrices and of the Wiener graph operator. In this work we present a comparative study of several Wiener-type descriptors computed for vertex- and edge-weighted molecular graphs, corresponding to organic compounds with heteroatoms and multiple bonds. The acute toxicities toward Tetrahymena pyriformis of 47 nitrobenzenes are modeled with multilinear regression equations, using as structural descriptors the hydrophobicity (corrected for ionization) and various Wiener-type indices, with better results than a comparative molecular field analysis model.

An algorithm for the calculation of the hyper-Wiener index of benzenoid hydrocarbons

An algorithm for the calculation of the hyper-Wiener index (WW) of benzenoid hydrocarbons (both cata- and pericondensed) is described, based on the consideration of pairs of elementary cuts of the corresponding benzenoid graph B. A pair of elementary cuts partitions the vertices of B into four classes. WW is expressed as a sum of terms of the form n11n22 + n12n21, each associated with a pair of elementary cuts; nrs, r, s = 1, 2 are the numbers of vertices in the respective four classes. The algorithm proposed enables a relatively easy calculation of WW, finding expressions for WW of homologous series of benzenoid hydrocarbons, and envisaging the relations between WW and molecular structure.