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.     

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.     

Expected distance based on random walks

By considering a graph as a network of resistances, Klein and Randi? (J Math Chem 12(1):81–95, 1993) proposed the definition of a distance measure. Indeed, if each edge of the graph represents a resistance of \(1 \varOmega \), the equivalent resistance of the graph between each pair of vertices may be used as a distance. Based upon random walks in graphs, Stephenson and Zelen (Soc Netw 11(1):1–37, 1989) built a computational model to find the probability that each edge is used. From a mathematical point of view, both articles are based upon exactly the same model and the link between random walks and the electrical representation was established by Newman (Soc Netw 27(1):39–54, 2005) when defining an alternative to Freeman’s (Sociometry 40:35–41, 1977, Soc Netw 1(3):215–239, 1979) betweenness centrality based upon random walks. In the present paper, the similitude between these two processes is exploited to propose a new random walks based distance measure that may be defined as the expected length of a walk between any pair of vertices. We call it the expected distance and we prove that it is actually a distance. From this new definition, the RW Index is proposed that sums the expected walks lengths between pairs of vertices exactly in the same way as the Wiener index sums the shortest paths distances or the Kirchhoff index sums the equivalent resistances. We compare the three indices and establish the vertex and the edge decompositions for both. We compute some value of the RW index for some families of graphs and conjecture the upper and lower bounds of the RW index.     

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.     

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.     

On walks in molecular graphs.

Walks in molecular graphs and their counts for a long time have found applications in theoretical chemistry. These are based on the fact that the (i, j)-entry of the kth power of the adjacency matrix is equal to the number of walks starting at vertex i, ending at vertex j, and having length k. In recent papers (refs 13, 18, 19) the numbers of all walks of length k, called molecular walk counts, mwc(k), and their sum from k = 1 to k = n - 1, called total walk count, twc, were proposed as quantities suitable for QSPR studies and capable of measuring the complexity of organic molecules. We now establish a few general properties of mwc's and twc among which are the linear dependence between the mwc's and linear correlations between the mwc's and twc, the spectral decomposition of mwc's, and various connections between the walk counts and the eigenvalues and eigenvectors of the molecular graph. We also characterize the graphs possessing minimal and maximal walk counts.     

On the centrality of vertices of molecular graphs

For acyclic systems the center of a graph has been known to be either a single vertex of two adjacent vertices, that is, an edge. It has not been quite clear how to extend the concept of graph center to polycyclic systems. Several approaches to the graph center of molecular graphs of polycyclic graphs have been proposed in the literature. In most cases alternative approaches, however, while being apparently equally plausible, gave the same results for many molecules, but occasionally they differ in their characterization of molecular center. In order to reduce the number of vertices that would qualify as forming the center of the graph, a hierarchy of rules have been considered in the search for graph centers. We reconsidered the problem of “the center of a graph” by using a novel concept of graph theory, the vertex “weights,” defined by counting the number of pairs of vertices at the same distance from the vertex considered. This approach gives often the same results for graph centers of acyclic graphs as the standard definition of graph center based on vertex eccentricities. However, in some cases when two nonequivalent vertices have been found as graph center, the novel approach can discriminate between the two. The same approach applies to cyclic graphs without additional rules to locate the vertex or vertices forming the center of polycyclic graphs, vertices referred to as central vertices of a graph. In addition, the novel vertex “weights,” in the case of acyclic, cyclic, and polycyclic graphs can be interpreted as vertex centralities, a measure for how close or distant vertices are from the center or central vertices of the graph. Besides illustrating the centralities of a number of smaller polycyclic graphs, we also report on several acyclic graphs showing the same centrality values of their vertices. © 2013 Wiley Periodicals, Inc.     

Graphical shapes: Seeing graphs of chemical curves and molecular surfaces

Interior seeing graphs of Jordan curves and two-dimensional closed surfaces are proposed for the characterisation or the shapes of chemical curves and surfaces. Some of the relevant properties of seeing graphs, in particular, of seeing trees, are described. The proposed applications of seeing graphs provide a molecular shape characterization by converting the continuum problem of a molecular contour surface, for example, Mat of an isodensity contour of molecular electronic charge distribution, into a discrete problem of a graph. The dependence of seeing graphs on be charge density contour value, a continuous parameter, is analysed. The family of seeing graphs occurring; within the chemically accessible range of charge density contour values is proposed for a computer-based analysis of molecular similarity. The general method is illustrated with the example of a detailed study on the family of seeing graphs of the electronic charge density of the ethene molecule.     

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.     

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.