Quantitative structure-property relationship (QSPR) modeling is crucial in cheminformatics and computational drug discovery for predicting the activity of compounds. Topological indices are a popular molecular descriptor in QSPR modeling due to their ability to concisely capture the structural and electronic properties of molecules. Here, we investigate the use of curvilinear regression models to analyze fibrates drug activity through topological indices, which modulate lipid metabolism and improve the lipid profile. Our QSPR approach predicts the physicochemical properties of fibrates based on degrees and distances from topological indices. Our results demonstrate that topological indices can enhance the accuracy of predicting physicochemical properties and biological activities of molecules, including drugs. We also conducted density functional theory (DFT) calculations on the investigated derivatives to gain insights into their optimized geometries and electronic properties, including symmetry. The use of topological indices in QSPR modeling, which considers the symmetry of molecules, shows significant potential in improving our understanding of the structural and electronic properties of compounds.

In this paper, we present a new approach that explores the application of graph energy variants in chemistry, specifically in the development of platinum anticancer drugs. While previous energy variants have been proposed without considering their direct relevance to chemistry, our study focuses on two key aspects. First, we investigate the correlation between seven degree-based and four distance-based topological indices and their corresponding energies in platinum anticancer drugs. Furthermore, we mathematically analyze the properties of these energies, establishing upper and lower bounds that can be generalized to other structures. Second, we examine the possibility of utilizing the energies of these topological indices as structural descriptors. Our research showcases promising results, suggesting potential improvements in the future manufacturing of anticancer drugs. In addition, we employ density functional theory calculations to optimize the molecular structures of platinum anticancer drugs and identify local reactive sites using Fukui functions. Quantum theory of atoms in molecules was carried out at the bond critical point , to reveal the nature of the intermolecular interactions in the investigated ten Pt anticancer drugs, especially, the nature of bonds between Pt atoms and their bond atoms. Overall, this study presents an innovative approach that bridges graph energies, topological indices, and DFT with the properties (physical and chemical) of platinum anticancer drugs, offering insights into their molecular properties and potential for enhanced drug design.

In this paper, we first introduce a new graph over an extension of semigroups and after that we study and characterize the spectral properties such as the diameter, girth, maximum and minimum degrees, domination number, chromatic number, clique number, degree sequence, irregularity index, and also perfectness for . Moreover, we state and prove some important known Zagreb indices on this new graph.

The aim of this paper is to illustrate how degree sequences may successfully be used over some graph products. Moreover, by taking into account the degree sequence, we will expose some new distinguishing results on special graph products. We will first consider the degree sequences of tensor and cartesian products of graphs and will obtain the omega invariant of them. After that we will conclude that the set of graphs forms an abelian semigroup in the case of tensor product whereas this same set is actually an abelian monoid in the case of cartesian product. As a consequence of these two operations, we also give a result on distributive law which would be important for future studies.

In [17], the authors established a new semigroup N as an extension of both Rees matrix and completely zero-simple semigroups. In this paper, by taking into account the Zappa-Sz\'{e}p product obtained by special subsemigroups of N, we will expose some new distinguishing theoretical results on this product.

In previous papers, it has been recently defined a new class of semigroups based on both Rees matrix and completely 0-simple semigroups. For this new structure, it has been introduced with certain basic properties and finiteness conditions. The main goal of the paper is to prove Green’s Theorem for by indicating the existence of Green’s lemma. We also study generalized Green’s relation over and present some new findings. In particular, we classify our semigroup associated with generalized Green’s relation by constructing good homomorphism.