Ghani Mersenne and Fermat Entropy for Some Dendrimers and Their Structure-Property Modeling

The mathematical language provided by various topological index types may be used to identify various properties of chemical components in a molecular structure. Within the framework of valency-based topological indices, we examine eight different dendrimer architectures. This chapter proposes two novel molecular descriptors, the Mersenne index and the Fermat index, by combining the essential notions of the Mersenne number and the Fermat number. The molecules with the highest Mersenne index and Fermat index values are found in the classes of some dendrimer families, including poly or propyl ether-imine dendrimer, polyethylene amido amine dendrimer, porphyrin dendrimer, zinc porphyrin dendrimer, porphyrin-cored dendrimer-I, porphyrin-cored dendrimer-II, porphyrin-cored dendrimer-III, and porphyrin-cored dendrimer-IV. The Fermat index differs from the Mersenne index in that it grows with the value of n, as the Fermat number is always increasing by increasing the value of n. Also, we show that the Mersenne index and the Fermat index are always members of the set of integers (Z) for any graph of chemical compounds.