I have developed PyTIE (Python Topological Indices Expressions) which is defined as the collections of Python packages such as PyTIE_D, PyTIE_DS, PyTIE_SMS_DE, and PyTIE_SMS_DSE, which are open-source software packages and cross-platform Python package designed to expedite the retrieval of results for mathematics, chemistry and chemical engineering researchers within constant time. This open-source tool extends its utility to chemistry and chemical engineering researchers with limited mathematical proficiency. PyTIE facilitates the loading of molecular graphs, specifying parameters such as minimum degree, maximum degree, and the number of vertex pairs (edge partitions). The degree-sum based edge partitions of a molecular graph also plays a vital role in predicting heat of formation and enthalpy of formation along with DFT techniques. It systematically computes expressions and numerical values for various topological indices, including degree-based and neighborhood degree-based indices, as well as Shannon's entropy, providing visual representations of the results. Emphasizing topological indices for Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR) analyses, PyTIE proves particularly relevant in these studies. Serving as a Python package, it seamlessly integrates with libraries such as NumPy, math and SymPy offering extensive options for data analysis. The efficiency of PyTIE is demonstrated through illustrative examples in various contexts.

PyTIE is defined as a collections of Python software packages,are PyTIE_D, PyTIE_DS, PyTIE_SMS_DE, and PyTIE_SMS_DSE are vailable as open source, offering optimized transforms and a modular library of optimization tools for solving linear chemical structure issues. Here PyTIE D, PyTIE DS, PyTIE SMS DE, and PyTIE SMS DSE stands follow Degree-based Python Topological Indices Expressions, Degree Sum-based Python Topological Indices Expressions, Degree and Entropy based Python Topological Indices Expressions for a Single Molecular Structure, Degree Sum and Entropy based Python Topological Indices Expressions for a Single Molecular Structure. Originally tailored for QSPR and QSAR applications, the software’s adaptability and the broad applicability of its mathematical techniques, emphasized in this study, indicate its efficiency and consistent time results in computational tasks.

PyTIE has been developed to generate expressions and numerical values for topological indices in single-parameter structures. This discovery holds significant value for chemists and mathematical chemists alike. It implies that the time complexity of PyTIE is O(1), aligning with the efficient computational principles outlined in the theory of time complexity within the molecular descriptors field. Comprising four distinct software packages, PyTIE encompasses PyTIE_D, PyTIE_DS, PyTIE_SMS_DE, and PyTIE_SMS_DSE. Each package houses a singular module dedicated to various types of molecular descriptors computation. The execution of these modules involves calling four functions: topoexpressd, topoexpressds, smstopoexpressd, and smstopoexpressds. These functions portray molecular graph representations in terms of minimum degree, maximum degree, and edge partitions, alongside methods for computing diverse graph invariants and topological indices expressions and numerical values.