Topological indices and QSPR analysis of some chemical structures applied for the treatment of heart patients

In this study, various beta-blocker drugs used for heart disease were analyzed, and their degree-based topological indices derived from the M-polynomial were calculated. Linear and quadratic regression analysis was used to obtain quantitative structure-property relationship models between the topological indices and eight the physicochemical properties of the drugs to determine their effectiveness. The results show that the harmonic index was the best predictor for boiling point, flashpoint, and enthalpy of vaporization, while the redefined third Zagreb index was effective for polarizability, molar refractivity, and molar volume. The inverse sum indeg index was found to be effective for molar refractivity, and the second modified Zagreb index was surface tension in linear regression models. In addition, the redefined third Zagreb index was the best predictor for polarizability and molar refractivity, while the second modified Zagreb index was effective for molar volume. The SDD index was found to be effective for surface tension in quadratic regression models.     

On reciprocal molecular topological index

We report some properties of the reciprocal molecular topological index RMTI of a connected graph, and, in particular, its relationship with the first Zagreb index M₁. We also derive the upper bounds for RMTI in terms of the number of vertices and the number of edges for various classes of graphs, including K _r+1 -free graphs with r ≥ 2, quadrangle-free graphs, and cacti. Additionally, we consider a Nordhaus-Gaddum-type result for RMTI.     

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.     

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.     

Sharp Bounds for the Second Zagreb Index of Unicyclic Graphs

The second Zagreb index M ₂(G) of a (molecule) graph G is the sum of the weights d(u)d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u. In this paper, we give sharp upper and lower bounds on the second Zagreb index of unicyclic graphs with n vertices and k pendant vertices. From which, and C _n have the maximum and minimum the second Zagreb index among all unicyclic graphs with n vertices, respectively.     

A novel approach to the use of graph theory in structure–activity relationship studies. Application to the qualitative evaluation of mutagenicity in a series of nonfused ring aromatic compounds

In this article we describe a novel approach to the application of graph theory in structure–activity relationship studies. An information–theoretical topological index for the vertices of a molecular graph has been used for the qualitative evaluation of the mutagenic activity of a series of nonfused ring aromatic compounds. The use of a vertex index contrasts with the conventional approach of using a topological index for the entire molecule. The idea is to identify regions, or substructures in the molecules (molecular graphs) which may be used to determine certain biological activity of chemical compounds. The results obtained in this paper indicate that the present approach is capable of classifying the mutagenic activity of the compounds under consideration and may find useful application in structure–activity relationship studies of diverse bioactive compounds.     

Discriminating tests of information and topological indices. Animals and trees

In this paper we consider 13 information and topological indices based on the distance in a molecular graph with respect to their discrimination power. The numerical results of discriminating tests on 3490528 trees up to 21 vertices are given. The indices of the highest sensitivity are listed on the set of 1528775 alkane trees. The discrimination powers of indices are also examined on the classes of 849285 hexagonal, 298382 square, and 295365 triangular simply connected animals. The first class of animals corresponds to the structural formulas of planar benzenoid hydrocarbons. The values of all indices were calculated for all classes of animals as well as for the united set of 1443032 animals. The inspection of the data indicates the great sensitivity of four information indices and one topological index.     

On topological indices of carbon nanocones and nanotori

Let G be a graph with n vertices and d _i is the degree of its ith vertex (d _i is the degree of v _i). In this article, we compute the redefined first Zagreb index, redefined second Zagreb index, redefined third Zagreb index, augmented Zagreb index of graphs carbon nanocones CNC _k[n], and nanotori [C₄C₆C₈(p,q)]. Also, compute the multiplicative redefined first Zagreb index, multiplicative redefined second Zagreb index, multiplicative redefined third Zagreb index, multiplicative augmented Zagreb index of carbon nanocones CNC _k[n], and nanotori [C₄C₆C₈(p,q)].     

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.