General zeroth-order randić index of zero divisor graph for the ring of integers modulo pn.

Abstract

A simple graph is a set of vertices, V(Γ) and a set of edges, E(Γ), where each edge 〈u − v〉 connects two different vertices u and v (there are no self-loops). In topological index, the general zeroth-order Randić index is defined as the sum of the degree of each vertex to the power of a ≠ 0. Given a ring R, let Γ(R) denote the graph whose vertex set is R, such that the distinct vertices a and b are adjacent provided that ab = 0 for the zero-divisor graph of a ring. In this paper, we present the general formula of the general zeroth-order Randić index of the zero-divisor graph for some commutative rings. The commutative ring in the scope of this research is the ring of integers modulo pn, where p is a prime number and n is a positive integer. The general zeroth-order Randić index is found for the cases a = 1, 2 and 3.