Structural interpretation of the topological index. 2. The molecular connectivity index, the Kappa index, and the atom-type E-State index

The structural interpretation is extended to the topological indices describing cyclic structures. Three representatives of the topological index, such as the molecular connectivity index, the Kappa index, and the atom-type E-State index, are interpreted by mining out, through projection pursuit combining with a number theory method generating uniformly distributed directions on unit sphere, the structural features hidden in the spaces spanned by the three series of indices individually. Some interesting results, which can hardly be found by individual index, are obtained from the multidimensional spaces by several topological indices. The results support quantitatively the former studies on the topological indices, and some new insights are obtained during the analysis. The combinations of several molecular connectivity indices describe mainly three general categories of molecular structure information, which include degree of branching, size, and degree of cyclicity. The cyclicity can also be coded by the combination of chi cluster and path/cluster indices. The Kappa shape indices encode, in combination, significant information on size, the degree of cyclicity, and the degree of centralization/separation in branching. The size, branch number, and cyclicity information has also been mined out to interpret atom-type E-State indices. The structural feature such as the number of quaternary atoms is searched out to be an important factor. The results indicate that the collinearity might be a serious problem in the applications of the topological indices.     

Four new topological indices based on the molecular path code

The sequence of all paths pi of lengths i = 1 to the maximum possible length in a hydrogen-depleted molecular graph (which sequence is also called the molecular path code) contains significant information on the molecular topology, and as such it is a reasonable choice to be selected as the basis of topological indices (TIs). Four new (or five partly new) TIs with progressively improved performance (judged by correctly reflecting branching, centricity, and cyclicity of graphs, ordering of alkanes, and low degeneracy) have been explored. (i) By summing the squares of all numbers in the sequence one obtains Sigmaipi(2), and by dividing this sum by one plus the cyclomatic number, a Quadratic TI is obtained: Q = Sigmaipi(2)/(mu+1). (ii) On summing the Square roots of all numbers in the sequence one obtains Sigmaipi(1/2), and by dividing this sum by one plus the cyclomatic number, the TI denoted by S is obtained: S = Sigmaipi(1/2)/(mu+1). (iii) On dividing terms in this sum by the corresponding topological distances, one obtains the Distance-reduced index D = Sigmai{pi(1/2)/[i(mu+1)]}. Two similar formulas define the next two indices, the first one with no square roots: (iv) distance-Attenuated index: A = Sigmai{pi/[i(mu + 1)]}; and (v) the last TI with two square roots: Path-count index: P = Sigmai{pi(1/2)/[i(1/2)(mu + 1)]}. These five TIs are compared for their degeneracy, ordering of alkanes, and performance in QSPR (for all alkanes with 3-12 carbon atoms and for all possible chemical cyclic or acyclic graphs with 4-6 carbon atoms) in correlations with six physical properties and one chemical property.     

Topological indices for molecular fragments and new graph invariants

Whereas the internal fragment topological index (IFTI) is calculated in the normal manner as for any molecule, the external fragment topological index (EFTI) is calculated so as to reflect the interaction between the excised fragment F and the remainder of the molecule (G-F). For selected topological indices (TIs), a survey of EFTI values, formulas and examples is presented. Some requirements as to the fragment indices are formulated and examined. In the discussion of the results, it is shown that for some TIs regularities exist in the dependence of EFTI values upon the branching of fragment F, or upon the marginal versus central position of the fragment F in the graph G. New vortex invariants can be computed as EFTI values for one-atom fragments over all graph vertices; by iteration, it is in principle possible to devise an infinite number of now vertex invariants.     

The overall Wiener index--a new tool for characterization of molecular topology.

Recently, the concept of overall connectivity of a graph G, TC(G), was introduced as the sum of vertex degrees of all subgraphs of G. The approach of more detailed characterization of molecular topology by accounting for all substructures is extended here to the concept of overall distance OW(G) of a graph G, defined as the sum of distances in all subgraphs of G, as well as the sum of eth-order terms, (e)OW(G), with e being the number of edges in the subgraph. Analytical expressions are presented for OW(G) of several basic classes of graphs. The overall distance is analyzed as a measure of topological complexity in acyclic and cyclic structures. The potential usefulness of the components of this generalized Wiener index in QSPR/QSAR is evaluated by its correlation with a number of properties of C3-C8 alkanes and by a favorable comparison with models based on molecular connectivity indices.     

Molecular cyclicity and centricity of polycyclic graphs. I. Cyclicity based on resistance distances or reciprocal distances

Rules for molecular cyclicity based on the global indices resulting from reciprocal distances (Harary number, H) or from resistance distances (Kirchhoff number, Kf) were tested in comparison with those elaborated earlier by means of the Wiener index, W. The Harary number and the Wiener number were found to match molecular cyclicity in an almost identical manner. The Kirchhoff number also generally follows cyclicity trends described previously. H is slightly less degenerate than is W, but Kf has practically no degeneracy in the graphs investigated here. Being much more discriminating than the Wiener number (i.e., practically nondegenerate), Kf allowed the formulation of new rules for systems formed from linearly condensed ribbons of even-membered rings with different sizes as well as for branched ribbons. The topological cyclicity patterns are thus reformulated in an extended basis, proceeding from three different graph metrics. © 1994 John Wiley & Sons, Inc.     

Novel Indices for the Topological Complexity of Molecules

Abstract

The development of molecular complexity measures is reviewed. Two novel sets of indices termed topological complexities are introduced proceeding from the idea that topological complexity increases with the overall connectivity of the molecular graph. The latter is assessed as the connectivity of all connected subgraphs in the molecular graph, including the graph itself. First-order, second-order, third-order, etc., topological complexities ⁱ TC are defined as the sum of the vertex degrees in the connected subgraphs with one, two, three, etc., edges, respectively. Zero-order complexity is also specified for the simplest subgraphs–the graph vertices. The overall topological complexity TC is then defined as the sum of the complexities of all orders. These new indices mirror the increase in complexity with the increase in the number of atoms and, at a constant number of atoms, with the increase in molecular branching and cyclicity. Topological complexities compare favorably to molecular connectivities of Kier and Hall, as demonstrated in detail for the classical QSPR test-the boiling points of alkanes. Related to the wide application of molecular connectivities to QSAR studies, a similar importance of the new indices is anticipated.     

Overall connectivities/topological complexities: a new powerful tool for QSPR/QSAR

Earlier attempts to assess the complexity of molecules are analyzed and summarized in a number of definitions of general and topological complexity. A concept which specifies topological complexity as overall connectivity, and generalizes the idea of molecular connectivities of Randic, Kier, and Hall, is presented. Two overall connectivity indices, TC and TC1, are defined as the connectivity (the sum of the vertex degrees) of all connected subgraphs in the molecular graph. The contributions to TC and TC1, which originate from all subgraphs having the same number of edges e, form two sets of eth-order overall connectivities, eTC and eTC1. The total number of subgraphs K is also analyzed as a complexity measure, and the vector of its eth-order components, eK, is examined as well. The TC, TC1, and K indices match very well the increase in molecular complexity with the increase in the number of atoms and, at a constant number of atoms, with the increased degree of branching and cyclicity of the molecular skeleton, as well as with the multiplicity of bonds and the presence of heteroatoms. The potential of the three sets of eth-order complexities for applications to QSPR was tested by the modeling of 10 alkane properties (boiling point, critical temperature, critical pressure, critical volume, molar volume, molecular refraction, heat of formation, heat of vaporization, heat of atomization, and surface tension), in parallel with Kier and Hall's molecular connectivity indices (k)chi. The topological complexity indices were shown to outperform molecular connectivity indices in 44 out of the 50 pairs of models compared, including all models with four and five parameters.     

Discrimination of isomeric structures using information theoretic topological indices

Three newly defined information theoretic topological indices, namely “degree complexity (I^d),” “graph vertex complexity (H^V),” and “graph distance complexity (H^D)” along with three other information indices have been used to study their discriminating power of 45 trees and 19 monocyclic graphs. It is found that the newly defined indices have satisfactory discriminating power while H^D has been found to be the only index to discriminate all the graphs studied.     

Structural features hidden in the degree distributions of topological graphs

on a basic element, vertex degree, of topological graphs, insights are obtained on the structural features hidden in the degree distributions (DD) of saturated hydrocarbons. The investigation shows that the cyclicity and branching features are mainly coded by the different mathematical characteristics of the degree distributions. Surprisingly, the center (or mathematical expectation) of a degree distribution corresponds to the cyclicity of a saturated hydrocarbon, and the dispersion (mean absolute deviation or MAD) around its center of a distribution is a measure of branching. The structural feature such as number of quaternary atoms is also mined out as a special case of branching. The cyclicity and branching information in the present work is with least human intervention, and an interesting thing is that the two features can be unified into the mathematical characteristics of a degree distribution. By the strict mathematical characteristics of a distribution, the structure features within the degree distributions (DD somer domains) are studied. The space spanned by the size (number of carbons), mathematical expectation, and MAD shows some enlightening results. The results also give a new idea on how to model the properties of diverse structures.     

Wiener indices of trees and monocyclic graphs with given bipartition

The Wiener index of a connected graph is defined as the sum of distances between all unordered pairs of its vertices. It has found various applications in chemical research. We determine the minimum and the maximum Wiener indices of trees with given bipartition and the minimum Wiener index of monocyclic graphs with given bipartition, respectively. We also characterize the graphs whose Wiener indices attain these values. © 2011 Wiley Periodicals, Inc. Int J Quantum Chem, 2012     

Common vertex matrix: A novel characterization of molecular graphs by counting

We present a novel matrix representation of graphs based on the count of equal‐distance common vertices to each pair of vertices in a graph. The element (i, j) of this matrix is defined as the number of vertices at the same distance from vertices (i, j). As illustrated on smaller alkanes, these novel matrices are very sensitive to molecular branching and the distribution of vertices in a graph. In particular, we show that ordered row sums of these novel matrices can facilitate solving graph isomorphism for acyclic graphs. This has been illustrated on all undecane isomers C₁₁H₂₄ having the same path counts (total of 25 molecules), on pair of graphs on 18 vertices having the same distance degree sequences (Slater's graphs), as well as two graphs on 21 vertices having identical several topological indices derived from information on distances between vertices. © 2013 Wiley Periodicals, Inc.     

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.     

Isomer discrimination by topological information approach

A comparative analysis of ten topological indices is made. No index is found to discriminate isomers uniquely. A combined topological index, named the superindex, consisting of a number of topological indices is proposed. Information theory is applied to express all components of the superindex on a common quantitative scale. The superindex is tested on the sample of 427 graphs consisting of all acyclic, monocyclic, and bicyclic graphs with 4–8 vertices.     

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     

Comparability graphs and molecular properties: IV. Generalizations and applications

The development of a recently proposed method for calculating molecular properties is outlined. The approach is based on the idea of constructing optimized compound samples for structure—property or structure-activity correlations by means of the so—called comparability graphs (CG) of isomeric compounds. A dynamic comparability principle is devised, proceeding from a series of standard molecular rearrangements described in graph—theoretical terms as rules on molecular branching and cyclicity. An extension of the approach is presented for both the construction of CG's and their combination for variable numbers of atoms. The method is applied to various physico-chemical properties, which are thus divided into three groups according to the degree to which they are conditioned by molecular topology. The Wiener topological index is shown to produce a highly linear correlation with the alkane critical densities and volumes, as well as with their heats and entropies of vaporization.Dedicated to the memory of Professor Oskar E. Polansky, a pioneer of chemical graph theory