Topological indices are invariant numerical quantities of a graph that give facts about the structure of graphs and are found to be very helpful in predicting the physical properties of aluminophosphates. The characteristics of aluminophosphates are similar to the characteristics of zeolites. Two examples of current applications are natural gas dehydration and humidity sensors. Researchers in chemistry and materials science are synthesizing new frameworks. There are many layers and holes in these substances. The technique used to predict natural behaviors among the physicochemical characteristics of chemical molecules in their basic network is known as topological indices. This study explains the vertex version of distance-based topological indices, the entropy of topological indices and their numerical analysis.

Fullerenes are an allotrope of carbon that create polyhedral cages. Their bond structures match the sole pentagon and hexagonal-faced planar cubic graphs. Several chemical properties of fullerenes can be studied using its graph structure. Any graph that models a particular molecular structure can be given a topological index or molecular descriptor. Based on the molecular descriptor, it is easy to assess mathematical data and conduct further research on a molecule's physicochemical characteristics. It is a beneficial technique to replace time-consuming, expensive, and labour-intensive laboratory experiments. Molecular descriptors play a significant role in molecular structural analysis by investigating quantitative structure-activity relationships (QSARs) and quantitative structure-property relationships (QSPRs). In this study, some novel degree-based topological indices, multiplicative degree-based topological indices, and entropy versions for fullerene cages $C_{36}{D}_{6h}$ and $C_{30}{D}_{5h}$ have been computed and derived formula for them. Also, we have obtained the numerical computation and graphical representation of degree-based topological indices and entropy values of $C_{36}{D}_{6h}$ and $C_{30}{D}_{5h}$. Understanding the topology of precursor fullerenes is undoubtedly aided by the results of our computations. GRAPHICAL ABSTRACT

This paper shows the technique for calculating the Wiener index of melamine cyanuric acid structure by dividing it into smaller vertex-weighted quotient graphs. Melamine cyanuric acid is a crystalline complex made up of a $1:1$ combination of melamine and cyanuric acid. One of the topological indices that have been investigated the most, both theoretically and in terms of practical applications, is the Wiener index. It has a strong relationship with boiling points and other quantities of molecules. The computational techniques decrease the computational cost by selecting such as the range of values for the Wiener index, the variety and the size of atoms and bonds, and the number of cycles. In this work, we have computed the Wiener index for melamine cyanuric acid structures and also have done a correlation between numerical values and chemical quantities.

Topological indices are invariant numerical fields of a graph that give facts about the structure of graphs and are found to be very helpful in predicting the physical properties of aluminophosphates. The characteristics of the aluminophosphates are similar to those of zeolites. Two examples of current applications are natural gas dehydration and humidity sensor. New frameworks are being synthesised by researchers in chemistry and materials science. There are many layers and holes in these substances. In this study, Vertex version of distance-based topological indices, the entropy of topological indices and their numerical analysis are explained.

A quantitative structural analysis was carried out to predict the physical properties of boric acid structure. A topological descriptor of chemical molecule structure is a numerical value that distinguishes between a base structure and its branching pattern in the knowledge of chemical, physical, and biological aspects of molecular structure. In modern chemistry, theoretical chemistry, pharmacology, toxicology, and environmental chemistry, a large number of numerical graph invariants (topological indices) have been established and used for correlation analysis. In this paper, edge-version of distance based topological descriptors like edge Wiener, edge Szeged, edge PI and vertex-edge Wiener are computed for the structure of boric acid graphite sheet and Theta, Omega, Sd and PI polynomials and their subsequent topological indices for boric acid graphite structure are quantified. Further, using Theta and Omega polynomials, we devise a new approach for calculating the PI and Sd indices.