The valence overall wiener index for unsaturated hydrocarbons

By replacing the distances between pairs of vertices with the relative distances, we define a novel valence overall Wiener index (VOW); the valence overall Wiener index extends the usefulness of the Wiener index and the overall Wiener index to unsaturated hydrocarbons.     

直链烷烃取代衍生物Wiener指数的简便计算方法

根据直链烷烃衍生物分子拓扑结构的特点,将直链烷烃衍生物拆分为由直链单元和取代基团几个部分构成,再根据每部分的拓扑结构特点给出相应的计算公式.从而提出了一个计算直链烷烃衍生物Wiener指数的简便方法,达到简化计算的目的.该方法简化了传统Wiener指数的计算方法,使Wiener指数的计算具有效率高、不易出错等优点,便于Wiener指数计算程序化,从而提高了Wiener指数的实用性.     

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     

How to compute the Wiener index of a graph

The Wiener index of a graphG is equal to the sum of distances between all pairs of vertices ofG. It is known that the Wiener index of a molecular graph correlates with certain physical and chemical properties of a molecule. In the mathematical literature, many good algorithms can be found to compute the distances in a graph, and these can easily be adapted for the calculation of the Wiener index. An algorithm that calculates the Wiener index of a tree in linear time is given. It improves an algorithm of Canfield, Robinson and Rouvray. The question remains: is there an algorithm for general graphs that would calculate the Wiener index without calculating the distance matrix? Another algorithm that calculates this index for an arbitrary graph is given.     

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     

Generalized Hosoya polynomials of hexagonal chains

Similar to the well-known Wiener index, Eu et al. [Int. J. Quantum Chem. 106 (2006) 423–435] introduced three families of topological indices including Schultz index and modified Schultz index, called generalized Wiener indices, and gave the similar formulae of generalized Wiener indices of hexagonal chains. They also mentioned three families of graph polynomials in x, called generalized Hosoya polynomials in contrast to the (standard) Hosoya polynomial, such that their first derivatives at x = 1 are equal to generalized Wiener indices. In this note we gave explicit analytical expressions for generalized Hosoya polynomials of hexagonal chains.     

The Wiener polynomial of a graph

The Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a q-analog of the Wiener index. We study some of the elementary properties of this polynomial and compute it for some common graphs. We then find a formula for the Wiener polynomial of a dendrimer, a certain highly regular tree of interest to chemists, and show that it is unimodal. Finally, we point out a connection with the Poincaré polynomial of a finite Coxeter group. © 1996 John Wiley & Sons, Inc.     

Graphs of unbranched hexagonal systems with equal values of the wiener index and different numbers of rings

Graphs of unbranched hexagonal systems consist of hexagonal rings connected with each other. Molecular graphs of unbranched polycyclic aromatic hydrocarbons serve as an example of graphs of this class. The Wiener index (or the Wiener number) of a graph is defined as the sum of distances between all pairs of its vertices. Necessary conditions for the existence of graphs with different numbers of hexagonal rings and equal values of the Wiener index are formulated, and examples of such graphs are presented.     

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.     

Topological estimation of proton-ligand formation constants of potential antitumour agents: Salicylhydroxamic acids

Proton-ligand formation constants of salicylhydroxamic acids (SHA) and their nuclear substituted derivatives have been estimated topologically using the normalized Wiener index, referred to as mean square Wiener index (Wms). Regression analysis of the data indicates that Wms can be used successfully for estimating and monitoring proton-ligand formation constants.     

The Graovac–Pisanski index of zig-zag tubulenes and the generalized cut method

The Graovac–Pisanski index, which is also called the modified Wiener index, was introduced in 1991 by Graovac and Pisanski. This variation of the classical Wiener index takes into account the symmetries of a graph. In 2016 Ghorbani and Klav?ar calculated this index by using the cut method, which we generalize in this paper. Moreover, we prove that in some cases the automorphism group of a zig-zag tubulene is isomorphic to the direct product of a dihedral group and a cyclic group. Finally, the closed formulas for the Graovac–Pisanski index of zig-zag tubulenes are calculated.     

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.     

On reciprocal reverse Wiener index

We report some properties, especially bounds for the reciprocal reverse Wiener index of a connected (molecular) graph. We find that the reciprocal reverse Wiener index possesses the minimum values for the complete graph in the class of n-vertex connected graphs and for the star in the class of n-vertex trees, and the maximum values for the complete graph with one edge deleted in the class of n-vertex connected graphs and for the tree obtained by attaching a pendant vertex to a pendant vertex of the star on n − 1 vertices in the class of n-vertex trees. These results are compared with those obtained for the ordinary Wiener index.     

The extremal structures of k-uniform unicyclic hypergraphs on Wiener index

The Wiener index of a connected k-uniform hypergraph is defined as the summation of distances between all pairs of vertices. We determine the unique k-uniform unicyclic hypergraphs with maximum and second maximum, minimum and second minimum Wiener indices, respectively.     

On the relation between W'/W index, hyper-Wiener index, and Wiener numbers

It is shown analytically that the W'/W index, the hyper-Wiener index, and the Wiener number are closely related graph-theoretical invariants for acyclic structures. A general analytical expression for the hyper-Wiener index of a tree is derived too.     

On general sum-connectivity index

We report some properties of the reverse degree distance of a connected (molecular) graph, and, in particular, its relationship with the first Zagreb index and Wiener index. We also show that the reverse degree distance satisfies the basic requirement for a branching index.     

On reverse degree distance

We report some properties of the reverse degree distance of a connected (molecular) graph, and, in particular, its relationship with the first Zagreb index and Wiener index. We also show that the reverse degree distance satisfies the basic requirement for a branching index.