On Certain Counting Polynomial of Titanium Dioxide Nanotubes

Abstract

In 1936, Polya introduced the concept of a counting polynomial in chemistry. However, the subject established little attention from chemists for some decades even though the spectra of the characteristic polynomial of graphs were considered extensively by numerical means in order to obtain the molecular orbitals of unsaturated hydrocarbons. Counting polynomial is a sequence representation of a topological stuff so that the exponents precise the magnitude of its partitions while the coefficients are correlated to the occurrence of these partitions. Counting polynomials play a vital role in topological description of bipartite structures as well as counts of equidistant and non-equidistant edges in graphs. Omega, Sadhana, PI polynomials are wide examples of counting polynomials. Mathematical chemistry is a division of abstract chemistry in which we debate and forecast the chemical structure by using mathematical models. Chemical graph theory is a subdivision of mathematical chemistry in which the structure of a chemical compound can be embodied by a labelled graph whose vertices are atoms and edges are covalent bonds between the atoms. We use graph theoretic technique in finding the counting polynomials of TiO2 nanotubes. Let ! be the molecular graph of TiO2. Then (!, !) = !!10!!+8!−2!−2 + (2! +1) !10!!+8!−2! + 2(! + 1)10!!+8!−2 In this paper, the omega, Sadhana and PI counting polynomials are studied. These polynomials are useful in determining the omega, Sadhana and PI topological indices which play an important role in studies of Quantitative structure-activity relationship (QSAR) and Quantitative structure-property relationship (QSPR) which are used to predict the biological activities and properties of chemical compounds. These counting polynomials play an important role in topological description of bipartite structures as well as counts equidistance and non-equidistance edges in graphs. Computing distancecounting polynomial is under investigation.