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     

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.     

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     

On the Wiener index of a graph

A modification of the Weiner index which properly takes into account the symmetry of a graph is proposed. The explicit formulae for the modified Wiener index of path, cycle, complete bipartite, cube and lattice graphs are derived and compared with their standard Wiener index.     

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.     

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.     

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.     

Wiener numbers of benzenoid hydrocarbons: two theorems

Two exact results on the Wiener numbers of catacondensed benzenoid hydrocarbons are obtained. All catacondensed isomers have Wiener numbers congruent modulo 8. Among unbranched catacondensed systems, helicenes have the minimum and linear polyacenes the maximum Wiener number.     

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.     

On the construction of the matching polynomial for unbranched catacondensed benzenoids

An algorithm for obtaining the matching polynomial of an arbitrary catacondensed unbranched benzenoid molecule is presented. It is based on multiplication of only three 5 x 5 transfer matrices I, J, K, and an appropriate terminal vector. The choice of the matrices is dictated by the history of the growth of the hexagonal “animals” (i.e., by the pattern of the successive fusions of the benzene rings). The approach also gives the number of Kekule valance structures, the count of conjugated circuits, the values of the topological index Z, and the characteristic polynomials.     

On the geometric–arithmetic index by decompositions-CMMSE

The concept of geometric–arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. There are many papers studying different kinds of indices (as Wiener, hyper–Wiener, detour, hyper–detour, Szeged, edge–Szeged, PI, vertex–PI and eccentric connectivity indices) under particular cases of decompositions. The main aim of this paper is to show that the computation of the geometric-arithmetic index of a graph G is essentially reduced to the computation of the geometric-arithmetic indices of the so-called primary subgraphs obtained by a general decomposition of G. Furthermore, using these results, we obtain formulas for the geometric-arithmetic indices of bridge graphs and other classes of graphs, like bouquet of graphs and circle graphs. These results are applied to the computation of the geometric-arithmetic index of Spiro chain of hexagons, polyphenylenes and polyethene.     

Generalized Wiener indices of zigzagging pentachains

The “pentachains” studied in this paper are graphs formed of concatenated 5-cycles. Explicit formulas are obtained for the Schultz and modified Schultz indices of these graphs, as well as for generalizations of these indices. In the process we give a more refined version of the procedure that earlier was reported for the ordinary Wiener index.     

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.     

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.     

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.     

Bounds on Harary index

In this paper, we obtain the lower and upper bounds on the Harary index of a connected graph (molecular graph), and, in particular, of a triangle- and quadrangle-free graphs in terms of the number of vertices, the number of edges and the diameter. We give the Nordhaus–Gaddum-type result for Harary index using the diameters of the graph and its complement. Moreover, we compare Harary index and reciprocal complementary Wiener number for graphs.