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.     

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.     

Analytical expressions for topological properties of polycyclic benzenoid networks

Quantitative structure‐activity and structure‐property relationships of complex polycyclic benzenoid networks require expressions for the topological properties of these networks. Structure‐based topological indices of these networks enable prediction of chemical properties and the bioactivities of these compounds through quantitative structure‐activity and structure‐property relationships methods. We consider a number of infinite convex benzenoid networks that include polyacene, parallelogram, trapezium, triangular, bitrapezium, and circumcorone series benzenoid networks. For all such networks, we compute analytical expressions for both vertex‐degree and edge‐based topological indices such as edge‐Wiener, vertex‐edge Wiener, vertex‐Szeged, edge‐Szeged, edge‐vertex Szeged, total‐Szeged, Padmakar‐Ivan, Schultz, Gutman, Randić, generalized Randić, reciprocal Randić, reduced reciprocal Randić, first Zagreb, second Zagreb, reduced second Zagreb, hyper Zagreb, augmented Zagreb, atom‐bond connectivity, harmonic, sum‐connectivity, and geometric‐arithmetic indices. In addition we have obtained expressions for these topological indices for 3 types of parallelogram‐like polycyclic benzenoid networks.     

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 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     

Smooth function topological structure descriptors based on graph-spectra

The spectral topological indices: energy and Estrada index of a graph are generalized for any smooth function. The smooth function indices are introduced for this purpose as well as, via positive definite square summable functions, the Shannon entropy of a graph. Some comparative values are given between linear chains and cycles.     

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.     

Statistical properties of carbon nanostructures

We look at modeling carbon nanostructures from a theoretical graph network view, where a graph has atoms at a vertex and links represent bonds. In this way, we can calculate standard statistical mechanics functions (entropy, enthalpy, and free energy) and matrix indices (Wiener index) of finite structures, such as fullerenes and carbon nanotubes. The Euclidean Wiener index (topographical index) is compared with its topological (standard) counterpart. For many of these parameters, the data have power law behavior, especially when plotted versus the number of bonds or the number of atoms. The number of bonds in a carbon nanotube is linear with the length of the nanotube, thus enabling us to calculate the heat of formation of capped (5,5) and (10,10) nanotubes. These properties are determined from atomic coordinates using MATLAB routines.     

Molecular modeling of wine polyphenols

Due to wide range of health effects of wine polyphenols, it is important to investigate the relationship between their structure and physical properties (quantitative structure–property relationship, QSPR). We have investigated linear, nonlinear (polynomial), and multiple linear relationships between given topological indices and molecular properties of main pharmacological active components of wine, such as molecular weight (MW), van der Waals volume (Vw), molar refractivity (MR), polar molecular surface area (PSA) and lipophilicity (log P). Partition coefficient (log P) was calculated using three different computer program (CLOGP, ALOGPS and MLOGP). The best models were achieved using the MLOGP program. Topological indices used for correlation analysis include: the Wiener index, W(G); connectivity indices, χ(G); the Balaban index, J(G); information-theoretic index, I(G); and the Schultz index, MTI(G). QSPR was performed on the set of 19 polyphenols and, particularly, on the group of phenolic acids, and on the group of flavonoids with resveratrol. The connectivity index has been successfully used for describing almost all parameters. Significant correlations were achieved between the Wiener index and van der Waals volume, as well as molecular weight.     

Hosoya polynomials of armchair open‐ended nanotubes

For a connected graph G we denote by d(G,k) the number of vertex pairs at distance k. The Hosoya polynomial of G is H(G,x) = ∑_k≥0 d(G,k)x^k. In this paper, analytical formulae for calculating the polynomials of armchair open‐ended nanotubes are given. Furthermore, the Wiener index, derived from the first derivative of the Hosoya polynomial in x = 1, and the hyper‐Wiener index, from one‐half of the second derivative of the Hosoya polynomial multiplied by x in x = 1, can be calculated. © 2006 Wiley Periodicals, Inc. Int J Quantum Chem, 2007     

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.     

Distance based indices in nanotubical graphs: part 1

Nanotubical structures are obtained by wrapping a hexagonal grid, and then possibly closing the tube with caps. We show that the size of a cap of a closed (k, l)-nanotube is bounded by a function that depends only on k and l, and that those extra vertices of the caps do not influence the obtained asymptotical value of the distance based indices considered here. Consequently the asymptotic values are the same for open and closed nanostructures. We also show that the asymptotic for Wiener index, Schultz index (also known as degree distance), and Gutman index for (all) nanotubical graphs of type (k, l) on n vertices are \(\frac{n^3}{6(k+l)}+O(n^2)\), \(\frac{3n^3}{2(k+l)}+O(n^2)\), and \(\frac{n^3}{k+l}+O(n^2)\), respectively. In all cases, the leading term depends on the circumference of the nanotubical graph, but not on its specific type. Thus, we conclude that these distance based topological indices seem not to be the most suitable for distinguishing nanotubes with the same circumference and of different type as far as the leading term is concerned.     

Wiener polynomials of some chemically interesting graphs

The chemist Harold Wiener found ??(G), the sum of distances between all pairs of vertices in a connected graph G, to be useful as a predictor of certain physical and chemical properties. The q‐analogue of ??, called the Wiener polynomial ??(G; q), is also useful, but it has few existing useful formulas. We will evaluate ??(G; q) for certain graphs G of chemical interest. © 2004 Wiley Periodicals, Inc. Int J Quantum Chem, 2004     

On closed derivation formulas of the Nirmala indices from the M-polynomial of a graph

A new category of polymeric materials is dendrimers. These are monodisperse macromolecules with a high branching level. The physical and chemical characteristics of these materials are strongly influenced by their structure. Dendrimers are ideal for a wide variety of biological and industrial applications due to their distinctive behaviour. Very recently, in 2021, the Nirmala index, and the first and second inverse Nirmala indices are proposed and calculated their values for four standard dendrimers. In this present article, we propose closed derivation formulas for finding the above variations of Nirmala indices of a graph in terms of its M-polynomial. We also determine the M-polynomials and their geometrical natures for some families of dendrimers. Finally, we compute the Nirmala indices for each of the considered dendrimers using the M-polynomial approach based on the proposed closed derivation formulas and get the same numeric results as originally calculated.