Trees with the same Topological Index JJ

Abstract

In the present paper we investigate the trees with the same JJ index (called JJ-equivalent trees). The topological index JJ is derived from the so called Wiener matrix introduced by Randic et al., in 1994. The Wiener matrix is constructed by generalizing the procedure of Wiener for evaluation of Wiener numbers in alkanes. From such matrices several novel structural invariants of potential interest in structure-property studies were obtained. These include higher Wiener numbers, Wiener sequences, and hyper-Wiener number, etc. The topological index JJ is constructed by considering the row sums of the Wiener matrix. A construction method for a class of JJ-equivalent trees is given. By using this method we construct the smallest pairs of non-isomorphic JJ-equivalent trees which have 18 vertices. In addition we report on groups of 3,4, and 6 non-isomorphic JJ-equivalent trees. The smallest such trees are of size 28 for triples and quadruples, and 34 for the group of 6 elements.     

The Wiener distance matrix for acyclic compounds and polymers

The relationship between the Wiener indices and the topological structures of alkanes is analyzed. The expressions for the Wiener distances between elements of these structures are derived, and the distance matrix is constructed for them; this matrix is naturally called the Wiener distance matrix. The expressions for the Wiener indices of polymers with units of arbitrary structure are obtained.     

A new approach for devising local graph invariants: Derived topological indices with low degeneracy and good correlation ability

A new approach is presented for obtaining graph invariants which have very high discriminating ability for different vertices within a graph. These invariants are obtained as the solution set (local invariant set, LOIS) of a system of linear equationsQ · X = R, whereQ is a topological matrix derived from the adjacency matrix of the graph, andR is a column vector which encodes either a topological property (vertex degree, number of vertices in the graph, distance sum) or a chemical property (atomic number). Twenty examples of LOOIs are given and their degeneracy and ordering ability of vertices is discussed. Interestingly, in some cases the ordering of vertices obtained by means of these invariants parallels closely the ordering from an entirely different procedure based on Hierarchically Ordered Extended Connectivities which was recently reported. New topological indices are easily constructed from LOISs. Excellent correlations are obtained for the boiling points and vaporization enthalpies of alkanesversus the topological index representing the sum of local vertex invariants. Les spectacular correlations with NMR chemical shifts, liquid phase density, partial molal volumes, motor octane numbers of alkanes or cavity surface areas of alcohols emphasize, however, the potential of this approach, which remains to be developed in the near future.     

Topographical distance matrices for porous arrays

The topographical Wiener index is calculated for two-dimensional graphs describing porous arrays, including bee honeycomb. For tiling in the plane, we model hexagonal, triangular, and square arrays and compare with topological formulas for the Wiener index derived from the distance matrix. The normalized Wiener indices of C₄, T₁₃, and O(4), for hexagonal, triangular, and square arrays are 0.993, 0.995, and 0.985, respectively, indicating that the arrays have smaller bond lengths near the center of the array, since these contribute more to the Wiener index. The normalized Perron root (the first eigenvalue, λ ₁), calculated from distance/distance matrices describes an order parameter, f = l₁/n{\phi=\lambda_1/n} , where f = 1{\phi= 1} for a linear graph and n is the order of the matrix. This parameter correlates with the convexity of the tessellations. The distributions of the normalized distances for nearest neighbor coordinates are determined from the porous arrays. The distributions range from normal to skewed to multimodal depending on the array. These results introduce some new calculations for 2D graphs of porous arrays.     

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.     

On structural interpretation of several distance related topological indices.

We consider the role that individual bonds play in bond-additivities in order to better understand the structural basis of various topological indices. In particular we consider indices closely related to the Wiener index (W) and the distance matrix and search for optimal weights of terminal and interior CC bonds in alkanes for a selection of physicochemical properties. It is interesting to note that different properties are associated with different relative roles of the exterior and the interior CC bonds.     

Computer generation of distance polynomials of graphs

A computer program is developed to compute distance polynomials of graphs containing up to 200 vertices. The code also computes the eigenvalues and the eigenvectors of the distance matrix. It requires as input only the neighborhood information from which the program constructs the distance matrix. The eigenvalues and eigenvectors are computed using the Givens-Householder method while the characteristic polynomials of the distance matrix are constructed using the codes developed by the author before. The newly developed codes are tested out on many graphs containing large numbers of vertices. It is shown that some cyclic isospectral graphs are differentiated by their distance polynomials although distance polynomials themselves are in general not unique structural invariants.     

Modification of the Wiener index 4

A novel topological index W(F) is defined by the matrices X, W, and L as W(F) = XWL. Where L is a column vector expressing the characteristic of vertices in the molecule; X is a row vector expressing the bonding characteristics between adjacent atoms; W is a reciprocal distance matrix. The topological index W(F), based on the distance-related matrix of a molecular graph, is used to code the structural environment of each atom type in a molecular graph. The good QSPR/QSAR models have been obtained for the properties such as standard formation enthalpy of inorganic compounds and methyl halides, retention indices of gas chromatography of multiple bond-containing hydrocarbons, aqueous solubility, and octanol/water partition of benzene halides. These models indicate that the idea of using multiple matrices to define the modified Wiener index is valid and successful.     

Calculation of retention indices by molecular topology. Chlorinated benzenes

A comparative study was undertaken to test the ability of several different topological indices to predict the retention indices of chlorinated benzenes on polar and non-polar stationary phases using both correlation coefficients and correctly predicted elution sequences as criteria of fit. The test was performed on three topological indices: connectivity indices, Wiener numbers, and Balaban indices. The regression analyses showed that the molecular connectivity model predicted the retention indices of chlorinated benzenes more successfully than either Wiener numbers or Balaban indices. The results also demonstrated that the major structural property controlling chromatographic behavior was the size of the chlorinated benzene. In addition, the use of the new non-empirical heteroatom parameterization scheme in the calculation of Wiener numbers and Balaban indices was successfully tested for the first time.     

Structure/property correlation of substituted compounds based on graph theory

The molecular graph of compounds containing heteroatoms is expressed by introducing a new parameter β into the adjacency matrix at the position of the carbon-heteroatom pairs. The Wiener equation is modified to correlate some physical properties (normal boiling points, critical constants and refractive index) of acyclic halogen derivatives, ethers, amines and alcohols with the graph invariants (modified Wiener number, 3-path number and 2-path number). The calculation yields successful results except for alcohols.     

From Wiener index to molecules

In this paper we present an algorithm for the generation of molecular graphs with a given value of the Wiener index. The high number of graphs for a given value of the Wiener index is reduced thanks to the application of a set of heuristics taking into account the structural characteristics of the molecules. The selection of parameters as the interval of values for the Wiener index, the diversity and occurrence of atoms and bonds, the size and number of cycles, and the presence of structural patterns guide the processing of the heuristics generating molecular graphs with a considerable saving in computational cost. The modularity in the design of the algorithm allows it to be used as a pattern for the development of other algorithms based on different topological invariants, which allow for its use in areas of interest, say as involving combinatorial databases and screening in chemical databases.     

Degeneracy of some matrix graph invariants

New matrices associated with graphs and induced global and local topological indices of molecular graphs were proposed recently by Diudea, Minailiuc and Balaban. These matrices in canonical form are matrix graph invariants. A combined degeneracy of such invariants is considered. For every case of degeneracy corresponding graphs are presented.