A novel definition of the overall hyper-wiener index for unsaturated hydrocarbons

By replacing the distances between pairs of vertices with the relative distances, we define a novel overall hyper-Wiener index (NOR); the novel overall hyper-Wiener index extends the usefulness of the hyper-Wiener index and the overall hyper-Wiener index to unsaturated hydrocarbons.     

Three methods for calculation of the hyper-Wiener index of molecular graphs

The hyper-Wiener index WW of a graph G is defined as WW(G) = (summation operator d (u, v)(2) + summation operator d (u, v))/2, where d (u, v) denotes the distance between the vertices u and v in the graph G and the summations run over all (unordered) pairs of vertices of G. We consider three different methods for calculating the hyper-Wiener index of molecular graphs: the cut method, the method of Hosoya polynomials, and the interpolation method. Along the way we obtain new closed-form expressions for the WW of linear phenylenes, cyclic phenylenes, poly(azulenes), and several families of periodic hexagonal chains. We also verify some previously known (but not mathematically proved) formulas.     

Condensed Extended Hyper-Wiener Index

According to the definitions of molecular connectivity and hyper-Wiener index, a novel set of hyper-Wiener indexes （Dn, ^mDn） were defined and named as condensed extended hyper-Wiener index, the potential usefulness of which in QSAR/QSPR is evaluated by its correlation with a number of C3-C8 alkanes as well as by a favorable comparison with models based on molecular connectivity index and overall Wiener index.     

Hyper-Wiener index for acyclic structures

The discriminating ability of the hyper-Wiener index (the ability to discriminate between isomers) is discussed on a class of acyclic structures (trees) including the molecular graphs of alkanes. For trees with 4 to 26 vertices, numerical data on the classes of trees with coincident indices were obtained. For classes with maximal power, the diagrams of the corresponding trees are presented. Using the index for processing chemical structural data is discussed. Translated fromZhurnal Strukturnoi Khimii, Vol. 40, No. 2, pp. 351–357, March–April, 1999.     

The Overall Hyper-Wiener Index

The concept of overall connectivity of a graph G was extended here to the definition of the overall hyper-Wiener index OWW(G) of a graph G, defined as the sum of the hyper-Wiener indexes in all subgraphs of G, as well as the sum of eth-order terms, ^e OWW(G), with e being the number of edges in the subgraph. The potential usefulness of the overall hyper-Wiener index in QSAR/QSPR is evaluated by its correlation with a number of properties of C3-C8 alkanes and cycloalkanes.     

GTI-space: the space of generalized topological indices

A new extension of the generalized topological indices (GTI) approach is carried out to represent “simple” and “composite” topological indices (TIs) in an unified way. This approach defines a GTI-space from which both simple and composite TIs represent particular subspaces. Accordingly, simple TIs such as Wiener, Balaban, Zagreb, Harary and Randić connectivity indices are expressed by means of the same GTI representation introduced for composite TIs such as hyper-Wiener, molecular topological index (MTI), Gutman index and reverse MTI. Using GTI-space approach we easily identify mathematical relations between some composite and simple indices, such as the relationship between hyper-Wiener and Wiener index and the relation between MTI and first Zagreb index. The relation of the GTI-space with the sub-structural cluster expansion of property/activity is also analysed and some routes for the applications of this approach to QSPR/QSAR are also given.     

Wiener, hyper-Wiener, detour and hyper-detour indices of bridge and chain graphs

Given a collection of connected graphs one may build bridge and chain graphs out of them. In this paper it is shown how the Wiener, hyper-Wiener, detour and hyper-detour indices for bridge and chain graphs are determined from the respective indices of the individual graphs. The results obtained are illustrated by some examples.     

A Novel Lu Index to QSPR Studies of Aldehydes and Ketones

A novel index based on the hyper-Wiener index, named Lu, was proposed. The relative bond length between two adjacent vertices in a molecular graph was taken into account in the definition of the Lu index. The usefulness of the new index in QSPR study was verified by its correlation with a number of organic compounds including aliphatic aldehydes and ketones. For each of the physical properties (the normal boiling points, molar refractions and gas heat capacities at 25°C), high quality QSPR models were obtained. The final models were validated to be statistically reliable using the leave-one-out cross validation and/or an external test set. The correlation coefficients (> 0.99) of all constructed models indicate the necessity of such an index, and show the potential of the Lu index for QSAR/QSPR studies.     

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.     

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.     

Retention index,connectivity index and Van der Waals' volume of alkanes (GLC)

Summary The relationships between retention index and Van der Waals' volume and between retention index and connectivity index have been studied for 58 different alkanes (C₁–C₉) on squalane. The correlation coefficient for the former is higher than for the latter. From these equations a linear relationship between Van der Waals' volume and connectivity index is obtained which indicates that the two parameters are equivalent. A simple method for calculating the Van der Waals' volume of alkanes is proposed.     

Molecular topology

General formulas for calculating the Wiener index (W) ² and the hyper-Wiener index (R)³ in spiro-graphs containing three- to six-membered rings are proposed. They are derived on the basis of Hosoya's formula⁴ and the Klein-Lukovitz-Gutman⁵ formula for evaluatingW andR, respectively, in cycle-containing graphs, by using the layer matrix of cardinality (LC). ⁶ An extension of the Wiener number, theW ^* number of Gutman⁷ is also evaluated for these spiro-graphs.For Part 23, see Ref. 1.Dedicated to Academician of the RAS N. S. Zefirov (on his 60th birthday).Translated fromIzvestiya Akademii Nauk. Seriya Khimicheskaya, No. 9, pp. 1674–1679, September, 1995.     

Hosoya polynomials of TUC4C8(S) nanotubes

The Hosoya polynomial of a chemical graph G is , where d _G (u, v) denotes the distance between vertices u and v. In this paper, we obtain analytical expressions for Hosoya polynomials of TUC₄C₈(S) nanotubes. Accordingly, the Wiener index, obtained by Diudea et al. (MATCH Commun. Math. Comput. Chem. 50, 133–144, (2004)), and the hyper-Wiener index are derived. This work is supported by the Fundamental Research Fund for Physics and Mathematic of Lanzhou University (Grant No. LZULL200809).     

Cluster analysis: significance,empty space,clustering tendency,non-uniformity. II--Empty Space index

The here presented Empty Space index (ES) evaluates the fraction of the information space without experimental points, i.e. the space where the distance from an experimental point is significantly larger than the mean distance between the experimental points themselves. ES can be used to eliminate the ambiguity of the some clustering indexes, that aim to evaluate the separation of the data set in clusters, but these clustering indexes are really a mixed measure of clustering, of empty space (the empty space does not necessarily correspond to the break between clusters) and of the degree of uniformity of the objects. The ES index can be used also to correct the MST index, the clustering index based on the distribution of edge lengths in the minimum spanning tree connecting the objects. The corrected MST index seems to be a reliable measure of the clustering degree.     

On geometric-arithmetic index

The geometric-arithmetic (GA) index is a newly proposed graph invariant in mathematical chemistry. We give the lower and upper bounds for GA index of molecular graphs using the numbers of vertices and edges. We also determine the n-vertex molecular trees with the minimum, the second and the third minimum, as well as the second and the third maximum GA indices.     

On acyclic molecular graphs with maximal Hosoya index, energy, and short diameter

The total number of matchings of a graph is defined as its Hosoya index. Conjugated and non-conjugated acyclic graphs that have maximal Hosoya index and short diameter are characterized in this paper, explicit expressions of the Hosoya indices of these extremal graphs are also presented.     

Augmented Zagreb index

Inspired by recent work on the atom-bond connectivity (ABC) index we propose here a new topological index, augmented Zagreb index (AZI). The tight upper and lower bounds for chemical trees are obtained. Moreover, it has been shown that among all trees the star has the minimum AZI value. Characterizing trees with maximal augmented Zagreb index remains an open problem for future research.