Schultz and Modified Schultz Polynomials of Chain from Alternating Hexagonal and Quadruple Rings

Abstract

Many topological indices are closely related to chemical and physical properties, especially types of chemical structures that are characterized by the forms of chains of special chemical structures including hexacyclic, pentagonal, and tetracyclic structures. In 1947, the first chemist to find a relationship between topological index is called the Wiener index which was named after the chemical scientist Harold Wiener. He introduced the Wiener index to find a relationship between physic and chemical properties of chemical structures of molecular graphs. Then, the Hosoya polynomial in chemistry was found by Haruo Hosoya in 1988, through which the Wiener index was found, by finding the derivative of this polynomial and then substituting for the value of the variable with one. Therefore, our aim in this paper was to talk about other topological indices called Schultz and modified Schultz indices with mentioning their polynomials and to find general formulas for each of them for an alternating chain of quaternary and hexagonal rings, which have some applications in chemistry. Also, a program was made using the Mathematica program to find the polynomials, indices, and sketches of them with respect to the Schultz distance. The first researcher to talk about the Schultz index was Harry Schultz in 1989 and the first to talk about the modified Schultz index were Sandi Klavˇzar and Ivan Gutman in 1997. Finally, many types of research are found Schultz and modified Schultz to Lots of graphs and operations defined on it.