GENERALIZED REDEFINED ZAGREB INDEX ON SOME NANOSTRUCTURES

Chemical graph theory is currently expanding the use of topological indices to numerically encode chemical structure. The prediction of the characteristics provided by the chemical structure of the molecule is a key feature of these topological indices. The concepts from graph theory are presented in a brief discussion of one of its many applications to chemistry, namely, the use of topological indices in quantitative structure-activity relationship (QSAR) studies and quantitative structure-property relationship (QSPR) studies. This study uses the M-polynomial approach, a newly discovered technique, to determine the topological indices of the medication fenofibrate. With the use of degree-based topological indices, we additionally construct a few novel degree based topological descriptors of fenofibrate structure using M-polynomial. When using M-polynomials in place of degree-based indices, the computation of the topological indices can be completed relatively quickly. The topological indices are also plotted. Using M-polynomial, we compute novel formulas for the modified first Zagreb index, modified second Zagreb index, first and second hyper Zagreb indices, SK index, S K 1 index, S K 2 index, modified Albertson index, redefined first Zagreb index, and degree-based topological indices.

A fuzzy graph is one of the versatile application tools in the field of mathematics, which allows the user to easily describe the fuzzy relation between any objects. The nature of fuzziness is favorable for any environment, which supports to predict the problem and solving it. Fuzzy graphs are beneficial to give more precision and flexibility to the system as compared to the classical model (i.e.,) crisp theory. A topological index is a numerical quantity for the structural graph of the molecule and it can be represented through Graph theory. Moreover, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. A lot of topological indices are available in chemical-graph theory and H. Wiener proposed the first index to estimate the boiling point of alkanes called ‘Wiener index’. Many topological indices exist only in the crisp but it’s new to the fuzzy graph environment. The main aim of this paper is to define the topological indices in fuzzy graphs. Here, indices defined in fuzzy graphs are Modified Wiener index, Hyper Wiener index, Schultz index, Gutman index, Zagreb indices, Harmonic index, and Randić index with illustrations. Bounds for some of the indices are proved. The algorithms for distance matrix and MWI are shown. Finally, the application of these indices is discussed.

In 1972, within a study of the structure-dependency of total π-electron energy (E), it was shown that E depends on the sum of squares of the vertex degrees of the molecular graph (later named first Zagreb index), and thus provides a measure of the branching of the carbon-atom skeleton. Topological indices are found to be very useful in chemistry, biochemistry and nanotechnology in isomer discrimination, structure–property relationship, structure-activity relationship and pharmaceutical drug design. In chemical graph theory, a topological index is a number related to a graph which is structurally invariant. One of the oldest most popular and extremely studied topological indices are well–known Zagreb indices. In a (molecular) graph G, the Zagreb topological index is equal to the sum of squares of the degrees of vertices of G and the Zagreb coindex is defined as the sum of a graph’s vertex degrees which is not adjacent. In this paper, we obtain the Zagreb coindex of four operations on graphs.

In this paper, we formulate the degree-based topological indices such as first Zagreb index, second Zagreb index, third Zagreb index, atom bond connectivity index, geometric-arithmetic (GA) index, general sum connectivity index, hyper-Zagreb index, augmented Zagreb index, sum connectivity index and general Randic index Rλ (℘), for λ = © −1, 1, − 1 2 , 1 2 ª for grape seed Proanthocyanidins network. By using above-mentioned topological indices, this study provides us different outcomes.

Graph theory had wide applications in developing in silico tools and it is widely used to calculate topological indices to establish structural activity relations of chemicals/compounds. However, usage of Zagreb indices with respect to natural compounds activity/toxicity prediction needs more attention. Many available online tools are using atom bond connectivity index (ABC Index), first and second Zagreb indices. The usage of the Hyper Zagreb index is very rare and using natural compounds is neglected. In this context, three types of Zagreb indices (first Zagreb index, second Zagreb index and hyper Zagreb index) were calculated to the selected chemical compounds of natural products and the relation between these indices and cytotoxicity of natural compounds were established. We have selected IC50 Values of the selected natural compounds in Hela cell lines as an index for cytotoxicity from the literature. The correlation of Zagreb indices and activity was performed using the R program, and we reached the conclusion that all indices correlate with the cytotoxicity of the studied compounds. This study acts as evidence to prove that, hyper Zagreb index correlates more with the cytotoxicity/activity of the studied natural compounds. Further studies using other Machine Learning tools to verify these findings will establish the importance of the hyper Zagreb index as one method to predict the toxicity of natural compounds.

A graph invariant is one of the tools of topological indices used in chemical graph theory. Recently, most of chemist study in Zagreb indices and Zagreb coindices related to the connectivity of atom in molecular graph. Represented as graph, G. In this paper, we study two families of Polyphenylene dendrimers, namely D 1[n] and D 2[n]. We also compute their topological indices, Zagreb indices and Zagreb coindices based on the concept of the line graphs of the subdivision graphs. Furthermore, we give closed analytical results of these indices for graphs D 1[n] and D 2[n].

A topological index is a number related to the atomic index that allows quantitative structure–action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials was set up in the substance diagram hypothesis in light of vertex degrees. These indices are valuable in the investigation of calming exercises of certain compound systems. In this paper, we computed the first and second Zagreb index, the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials of the line graph of wheel and ladder graphs by utilizing the idea of subdivision.

A graph in which atoms are taken as vertices and bonds can be shown by edges is called a molecular graph. For such graphs we can find the indices showing their bioactivity as well as their physio-chemical properties. Industry is flourishing nowadays just because of the interdisciplinary study of different subjects. This gives us a way to understand the application of different subjects. Chemical graph theory is a mixture of chemistry and mathematics both play an important role in chemical graph theory. Chemistry provides a chemical compound and graph theory transform this chemical compound into a molecular graph which further studied by different aspects such as topological indices. In this article, we computed generalized Randić, general Zagreb, general sum-connectivity, ABC, GA, and indices of line graphs of the and nanotubes.”

The quantitative specifications which are used to narrate the atomic topology of graphs are usually entitled as topological indices in theoretical Chemistry. The features of physical and chemical properties like melting point, entropy, boiling point, energy generation and enthalpy of vaporization of chemical compounds can be estimated by means of these topological indices. The field of Graph theory has a remarkable application of linking certain graphs with many types of topological indices. In this paper we compute first and second Zagreb indices, Randi´c index, sum-connectivity index, harmonic index, inverse sum indeg index, modified first and second Zagreb indices and first and second hyper Zagreb indices of Triangular snake graph and alternate Triangular snake graph.

Non-Kekule benzenoid structures are molecular systems that cannot be represented by classical Kekule resonance forms with alternating single and double bonds. These compounds often possess unpaired electrons and exhibit distinct electronic characteristics, including diradical and polyradical behavior. In this study, we analyze the structural and topological properties of selected non-Kekule benzenoid systems by computing various molecular descriptors, including the Atom-Bond Connectivity (ABC) index, Geometric-Arithmetic (GA) index, Randić index, First and Second Zagreb indices, Hyper Zagreb index and the Forgotten index. These topological indices offer valuable insights into the connectivity and branching patterns of the molecules, contributing to a deeper understanding of their structural complexity. The results may serve as a foundation for future investigations into the potential applications of these systems in fields such as materials science and molecular design.

Extremal unicyclic and bicyclic graphs with respect to the [formula omitted]-index

Mathematical chemistry studies the chemical structure of molecules. Topological indices are numerical values which associates the chemical structure with the physical and chemical properties. Multiplicative Universal Zagreb indices are generalized degree-based topological indices which gave rise to several indices like first and second multiplicative, Zagreb and hyper-Zagreb indices, multiplicative sum and product conductivity indices. This research is designed to study the first and second multiplicative Universal Zagreb indices of carbon nanostructures (carbon nanosheet, nanotube and nanotorus) of the same chemical formula [Formula: see text] We also depict the values of first and second multiplicative, Zagreb and hyper-Zagreb indices as well as multiplicative sum and product connectivity indices for the mentioned structures. The graphical comparison for each of the multiplicative Zagreb indices is presented for all the carbon nanostructures with the same chemical formula [Formula: see text]

This study investigates the efficacy of eight multiplicative degree-based and three classical degree-based topological indices in Quantitative Structure-Property Relationship (QSPR) models for predicting critical physicochemical properties of 38 antidepressant drugs. Molecular structures of compounds including Bupropion, Amitriptyline, and Fluoxetine were translated into numerical descriptors using indices such as Multiplicative Sum Zagreb, Multiplicative Sombor, and the First Zagreb index. These descriptors were integrated with machine learning algorithms: Random Forest, XGBoost, and linear regression to forecast boiling points, melting points, critical temperature, critical volume, and molar refractivity. Results revealed that the XGBoost algorithm significantly outperformed other methods, achieving superior predictive accuracy with the lowest error metrics (e.g., for boiling point: MAE = 8.60, RMSE = 12.40, r^2= 0.995). Among the topological indices, the First Zagreb index (M_1(G)) emerged as the most robust descriptor, demonstrating the strongest correlations with key properties (e.g., r = 0.932 with critical volume, r = 0.938 with molar refractivity). Linear regression models further confirmed the significance of M_1(G) and other indices, with high statistical significance (p < 0.05) in most cases. This interdisciplinary approach demonstrates the potent synergy of graph-theoretic indices and advanced machine learning in pharmaceutical research, offering a powerful strategy to accelerate drug discovery and optimize the design of novel therapeutic agents.

A topological index of graph G is a numerical parameter related to G, which characterizes its topology and is preserved under isomorphism of graphs. Properties of the chemical compounds and topological indices are correlated. In this paper we will compute M-polynomial, first and second Zagreb polynomials and forgotten polynomial for the Cartesian Product of a path and a complete bipartite graph for all values of n and m. From the M-polynomial, we will compute many degree-based topological indices such that general Randic index, inverse Randic index, first and second Zagreb index, modified Zagreb index, Symmetric division index, Inverse sum index augmented Zagreb index and harmonic index for the Cartesian Product of a path and a complete bipartite graph. Also, we will compute the hyper- Zagreb index, the first and second multiple Zagreb index and forgotten index for the Cartesian Product of a path and a complete bipartite graph.

Chemical graph theory is a sub-branch of mathematical chemistry, assuming each atom of a molecule is a vertex and each bond between atoms as an edge. Owing to this theory, it is possible to avoid the difficulties of chemical analysis because many of the chemical properties of molecules can be determined and analyzed via topological indices. Due to these parameters, it is possible to determine the physicochemical properties, biological activities, environmental behaviours and spectral properties of molecules. Nowadays, studies on the zero divisor graph of Z_n via topological indices is a trending field in spectral graph theory. For a commutative ring R with identity, the prime ideal sum graph of R is a graph whose vertices are nonzero proper ideals of R and two distinctvertices I and J are adjacent if and only if I+J is a prime ideal of R. In this study the forgotten topological index and Wiener index of the prime ideal sum graph of Z_n are calculated for n=p^α,pq,p^2 q,p^2 q^2,pqr,p^3 q,p^2 qr,pqrs where p,q,r and s are distinct primes and a Sage math code is developed for designing graph and computing the indices. In the light of this study, it is possible to handle the other topological descriptors for computing and developing new algorithms for next studies and to study some spectrum and graph energies of certain finite rings with respect to PIS-graph easily.