Topological indices are numerical parameters used to study the physical and chemical properties of compounds. In quantitative structure–activity relationship QSARs, topological indices correlate the biological activity of compounds with their physical properties like boiling point, stability, melting point, distortion, and strain energy etc. In this paper, we determined the M-polynomials of the crystallographic structure of the molecules Cu₂O and T_iF₂ [p,q,r]. Then we derived closed formulas for some well-known topological indices using calculus. In the end, we used Maple 15 to plot surfaces associated with the topological indices of Cu₂O and T_iF₂ [p,q,r].

Numerous studies demonstrate a strong intrinsic relationship between the boiling and melting temperatures, among other chemical properties, of chemical compounds and pharmaceutical and their molecular structures. Using topological indices, researchers can learn more about the physical traits, chemical stability, and bioactivities of these chemical molecular structures. Topological indices on the chemical structure of chemical materials and drugs are investigated in order to make up for the absence of chemical experiments and provide a theoretical basis for the manufacture of medications and chemical materials. According to well-known degree-based topological indices, the chemical structures of drugs used to treat hepatitis (A, B, C, D, and E) are assessed in this study. The atoms are thought of as the vertices of a graph, and the borders that separate them are thought of as the edges. Using degree-based topological indices, a quantitative structure-property relationship (QSPR) investigation was conducted to predict the physical properties of 16 hepatitis medications. These topological indices link the chemical structure to specific physical characteristics, such as the surface tension of hepatitis medication molecules and molecular weight, enthalpy, boiling point, density, vapor pressure, and logP. Using their molecular structures, the study's drugs are represented as molecular graphs, and 14 topological indices are computed.

Data protection is regarded as one of the biggest issues facing companies that have been using public data for a long time. Numerous encryption techniques have been used to address these issues and safeguard data from any malicious attempts and assaults. A substitution box (S-box) is the basic component for modern block ciphers, which helps to ensure robust security of plain data while encryption and permits its lawful decipherment. The goal of this paper is to recommend an effective, original, and straightforward technique for the creation of robust S-boxes. A sample S-box is generated in the proposed work using the word “UNITY” but other words can also be used to generate many powerful S-boxes. The ASCII code is used to translate the word “UNITY” into binary form, after which a distinct matrix is constructed for each character of the word. In the next phase, a linear fractional transformation is constructed using these matrices, which is then utilized to generate the S-box. The constructed S-box was then evaluated against typical security standards to support its high cryptographic authority. The generated S-box's statistical and algebraic resilience is demonstrated by its very low linear probability and differential probability scores of 0.125 and 0.039, respectively, and a high non-linearity score of 111.5. To evaluate the effectiveness of the image encryption scheme, digital images are encoded using the created S-box. The performance and comparative research demonstrate that the suggested S-box is a real candidate for applications in the field of image encryption and has a stronger performance base.