Graph theory received a great attention in chemical graph theory which is a branch of mathematical chemistry to model a molecule through a graph where vertices are considered as the atoms of the molecule and edges are considered as the bonds between them. A topological index or molecular descriptor of a graph is a numeric quantity obtained mathematically which characterizes its topology and it is invariant under graph isomorphisms. In chemical graph theory, various topological indices play important role in predicting physicochemical properties of chemical compounds. In this paper, we compute the general Zagreb index of some silicate networks and hence obtain some degree based topological indices such as Zagreb indices, forgotten topological index, redefined Zagreb index, general first Zagreb index, general Randić index and symmetric division deg index of silicate networks.

The algebraic polynomial plays a significant role in mathematical chemistry to compute the exact expressions of distance-based, degree-distance-based, and degree-based topological indices. The topological index is utilized as a significant tool in the study of the quantitative structure activity relationship (QSAR) and quantitative structures property relationship (QSPR) which correlate a molecular structure to its different properties and activities. Graphs containing finite commutative rings have wide applications in robotics, information and communication theory, elliptic curve cryptography, physics, and statistics. In this article, the topological indices of the total graph T ℤ n n ∈ ℤ + , the zero divisor graph Γ ℤ r n ( r is prime, n ∈ ℤ + ), and the zero divisor graph Γ ℤ r × ℤ s × ℤ t ( r , s , t are primes) are computed using some algebraic polynomials.

The concept of graph spectra can be thought of as an approach to use linear algebra including, in particular, the well developed theory of matrices for unlocking a thousand secrets about graph theory and its applications. A novel neighborhood degree sum based matrix is proposed as a modification of classical adjacency matrix. Using the spectrum of this matrix, a graph energy and its Estrada index are introduced, and their role as a molecular structural descriptor in chemical graph theory is investigated. An algorithm is designed to make the computation of the energy and its Estrada index convenient. The relationship between the recently proposed matrix and its associated graph invariant is studied using the spectral moment. Several sharp bounds for spectral radius, energy, and Estrada index are computed, and the corresponding extremal graphs are characterized. The integral representation of the energy is also reported.

In this paper, we mainly focused on some well-known degree-based topological indices. How they can help in QSPR studies is the main objective behind this study. We tested their predictive ability of some physicochemical properties of polycyclic aromatic hydrocarbons. With the help of this study, we predicted boiling point, entropy, acentric factor, octanol-water partition coefficient, enthalpy of formation and Kovats retention index of some benzenoid hydrocarbons.

The novel coronavirus disease 2019 (COVID-19) emerged in Wuhan, China, and has spread rapidly to nearly every part of the world. Unfortunately, no drug or vaccine has been accepted for the treatment of this pandemic. Researchers have established the efficacy of some existing antiviral drugs to control COVID-19 in vitro. Some of them are remdesivir (GS-5734), chloroquine, hydroxychloroquine, theaflavin. Topological indices are mathematical interpretations of a molecule generated by an algorithm implemented to a given molecular representation. Topological indices are used to model different physicochemical properties and biological activities of chemical compounds. In this work, some degree-based and neighborhood degree sum-based topological indices are investigated for the aforesaid antiviral drugs using polynomial approach. The results obtained can aid in the design of new medicine for the treatment of COVID-19.

In this paper, we mainly concern with some neighborhood degree-based multiplicative topological indices that are introduced by some eminent researchers in the past three years. According to the suggestion of the “International Academy of Mathematical Chemistry,” we tested their predictive ability of physicochemical properties of molecules with the help of a data set of octane isomers. Here, we also modified the general fifth multiplicative Zagreb index as a new one and computed this index for some line graphs of subdivision graphs of some union graphs of hexagons. Hence, computed some other multiplicative topological indices with the help of this index.

A molecular graph is hydrogen deleted simple connected graph in which vertices and edges are represented by atoms and chemical bonds, respectively. Topological indices are numerical parameters of a molecular graph which characterize its topology and are usually graph invariant. In Mathematical chemistry, topological descriptors play an important role in modeling different physical and chemical activities of molecules. In this study, the generalized Zagreb index for three types of carbon nanotubes is computed. By putting some particular values to the parameters, some important degree-based topological indices are also derived.

A topological index is a numerical quantity connected with a graph describing the molecular topology of the graph. It can predict different physicochemical properties such as boiling point, entropy, acentric factor etc. of chemical compounds. Dendrimers are highly branched nanostructures that are regarded as a building block in nanotechnology having wide applications. In this paper, multiplicative degree-based topological indices are computed for some nanostar dendrimers. The derived results have the potential for implementation in the chemical, biological, and pharmaceutical sciences.

A graph is defined by a set of objects, named as vertices, some of which are connected by links, known as edges. The term network and graph are same words and are used based on their domain of applications. Now a days, networking research received a great attention in the field of electrical and electronic engineering. There are various types of networks, which are classified to their connection types, its architecture and topology. A topological index of a graph structure or a network is a numeric quantity derived from that graph structure or network by mathematically which is correlated with various structural properties. In this paper, we computed some multiplicative version topological indices of block shift and hierarchical hypercube networks.

The graph energy, closely related to the total pi-electron energy of conjugated hydrocarbon molecules, is the sum of absolute characteristic values of adjacency matrix of the graph. The journey of such molecular descriptors is proceeding through the various graph energies based on different graph theoretical matrices. In this work, some novel graph energies based on some new graph theoretical matrices are presented. The applicability of them as molecular descriptors is discussed with some quantitative structure property relationship models. Also isomer discrimination ability of these energies are illustrated. In addition, the energies are calculated for some special graphs and some bounds of the energies are established.

Topological index is a connection between the chemical structure and the real number that remains invariant under graph isomorphism. In structure–property and structure–activity modeling, topological indices are considered as essential molecular descriptors to predict different physicochemical properties of molecule. Dendrimers are considered to be the most significant, commercially accessible basic components in nanotechnology. In this report, some neighborhood degree sum-based molecular descriptors are obtained for the fractal tree and the Cayley tree dendrimers. Neighborhood M-polynomial yields a family of topological indices for a molecular graph in less time compared to the usual computation from their definitions. Some indices are obtained using neighborhood M-polynomial approach. In addition, some multiplicative neighborhood degree sum-based molecular descriptors are evaluated for fractal and Cayley tree dendrimers. The graphical representations of the outcomes are presented. A comparative study of the findings with some well-known degree-based indices is performed. Usefulness of the descriptors in modeling different properties and activities is discussed.

BACKGROUND:Topological index is a numerical molecular descriptor that plays an important role in structure-property/structure-activity modeling. A large number of works on multiplicative degree based indices have been developed. However, no attention is paid to investigating their chemical significance. Investigation of the chemical importance of such indices is needed. The computation of topological indices for different chemical structures and networks is a current topic of interest in mathematical chemistry.OBJECTIVE:The objective of the present work is to examine the usefulness of the multiplicative degree based indices in quantitative structure property/activity relationship modeling. In addition, we intend to compute the indices for some anti-COVID-19 chemicals.MATERIALS AND METHODS:The regression analysis for octane data set is performed using MATLAB and Excel to check the predictability of the indices. The sensitivity test is conducted to examine the isomer discrimination ability. To study the indices for chemical structures preventing COVID-19, different combinatorial computation methods are utilized.RESULTS AND DISCUSSION:The regression models governing the structural dependence of different properties and activities are derived. The supremacy of the indices as useful molecular descriptors compared to some well-known and most used descriptors is established. Explicit expressions of the indices for hydroxychloroquine, remdesivir (GS-5734) and theaflavin are obtained.CONCLUSION:As the indices are shown to have remarkable efficiency in quantitative structure property/activity relationship modeling and isomer discrimination, the outcomes can predict different properties and activities of the chemicals under consideration.

The neighborhood M-polynomial is effective in recovering neighborhood degree sum based topological indices that predict different physicochemical properties and biological activities of molecular structures. Topological indices can transform the information found in molecular graphs and networks into numerical characteristics and thus make a major contribution to the study of structureproperty and structure-activity relationships. In this work, the neighborhood M-polynomial of the paraline graph of some convex polytopes is obtained. From the neighborhood M-polynomial, some neighborhood degree-based topological indices are recovered. Applications of the work are described. In addition, a quantitative and graphical comparison is made.

The neighborhood M-polynomial is effective in recovering neighborhood degree sum based topological indices that predict different physical, chemical and biological characteristics of material under investigation. In this work, the neighborhood M-polynomial of Titania nanotube TiO2 and the crystallographic structure of TiF2 are obtained. From the neighborhood M-polynomial, some neighborhood degree sum based topological indices are recovered. Effect of oxygen vacancies on outcomes is discussed. A comparative study among the findings and some well-established degree-based indices is performed.

Topological index is a numerical value associated with a chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. In this work, some new indices based on neighborhood degree sum of nodes are proposed. To make the computation of the novel indices convenient, an algorithm is designed. Quantitative structure property relationship (QSPR) study is a good statistical method for investigating drug activity or binding mode for different receptors. QSPR analysis of the newly introduced indices is studied here which reveals their predicting power. A comparative study of the novel indices with some well-known and mostly used indices in structure-property modelling and isomer discrimination is performed. Some mathematical properties of these indices are also discussed here.

Topological indices are numerical values associated with chemical constitution describing the structures of chemical compounds and helping to predict different physicochemical properties. In this report, some newly designed topological descriptors, namely, neighborhood Zagreb index (M-N), neighborhood version of Forgotten topological index (F-N), modified neighborhood version of Forgotten topological index (F-N*), neighborhood version of second Zagreb index (M-2*), neighborhood version of hyper Zagreb index (HMN) are obtained for the TURC4C8(S), armchair nanotube TUAC(6), V-phenylenic nanotube V PH X[m, n] and V-phenylenic nanotori VPHY[m, n].

In quantitative structure property relationship analysis (QSPR) and quantitative structure property relationship analysis (QSAR) the correlation between different properties/activities and molecular structure of chemical compounds is investigated which is helpful in drug design. Topological index is an useful tool to predict different physical and chemical properties of molecule by collecting information from the molecular graph. In this article, multiplicative degree based topological indices are obtained for some chemical structures widely used in drug design, especially in anticancer drug discovery. To visualize the indices, the results are interpreted graphically.

Topological indices are useful in QSAR/QSPR studies for modeling biological and physiochemical properties of molecules. The neighborhood Zagreb index (MN) is a novel topological index having good correlations with some physiochemical properties. For a simple connected graph G, the neighborhood Zagreb index is the totality of square of δG(v) over the vertex set, where δG(v) is the total count of degrees of all neighbors of v in G. In this report, some bounds are established for the neighborhood Zagreb index. Some explicit expressions of the index for some graph operations are also computed, which are used to obtain the index for some chemically significant molecular graphs.

A graph is a mathematical model used to predict the topology of a given system. In chemical graph theory, a graph is designed by considering atoms as vertices and edges as bonds between atoms of a particular molecule. A topological index or molecular structure descriptor is a numeric quantity associated with the chemical constitution which correlated with various physiochemical properties of the chemical structure. In this paper, we study the [Formula: see text]-Zagreb index of line graphs of the subdivision graphs of some chemical structures.

The Zagreb indices for a graph are defined as sum of the square of the degrees of all the vertices of the graph and sum of the product of degrees of all the pair of adjacent vertices of that graph. In this study, the Zagreb indices of two recently introduced graph operations, called double join of graphs based on the total graph and double corona of graphs based on the total graph are computed in terms of different topological indices of their factor graphs and hence some application of the derived results are also discussed.