Distances of zero-divisor type graphs for some rings of integers modulo n

Abstract

The zero-divisor graph of a ring is a graph whose vertices are the collection of zero-divisors of the ring, with two distinct vertices, x and y are connected by an edge if and only if xy=0. Meanwhile, a zero-divisor type graph is a compression of the zero-divisor graph by partitioning the vertices. For ring of integers modulo n, the zero-divisor type graph of ℤn denoted as ΓT(ℤn) is defined as graph with vertex set contains Td, where d is a nontrivial divisor of n and two distinct vertices, Tdi and Tdj are adjacent, if di · dj = 0. The zero-divisor type graph of ℤn is very useful in order to determine the perfectness of zero-divisor graph of ℤn where the graph concerns on vertices of the graph. The objective of this article is to compute the distances including the Wiener index and mean distance of the zero-divisor type graph of ℤn, ΓTℤn for some values of n. The Wiener index and mean distance of zero-divisor type graph of ℤn have been found to be constant for each factorization of n.