Wiener and Harary indices of the generalized zero-divisor graph

Abstract

Let R be a commutative ring, and let Z∗(R) denote the set of all non-zero zero-divisors in R. The generalized zero-divisor graph of R, denoted by Γ′(R), is a simple (undirected) graph with the vertex set Z∗(R), and two distinct vertices x and y are adjacent in Γ′(R) if and only if xny = 0 or ynx = 0, for some positive integer n. The Wiener and Harary indices are topological indices widely used in chemistry to analyze the structures of molecules. In this paper, we determine the Wiener and Harary indices of Γ′(Zn) for n = pq, p2q, p2q2, where p, q are distinct primes.