STRUCTURE OF ZERO-DIVISOR GRAPHS ASSOCIATED TO RING OF INTEGER MODULO n

For a commutative ring $R$ with identity $1\neq 0$, let $Z^{*}(R)=Z(R)\setminus \lbrace 0\rbrace$ be the set of non-zero zero-divisors of $R$, where $Z(R)$ is the set of all zero-divisors of $R$. The zero-divisor graph of $R$, denoted by $\Gamma(R)$, is a simple graph whose vertex set is $Z^{*}(R)=Z(R)\setminus \{0\}$ and two vertices of $ Z^*(R)$ are adjacent if and only if their product is $ 0 $. In this article, we find the structure of the zero-divisor graphs $ \Gamma(\mathbb{Z}_{n}) $, for $n=p^{N_1}q^{N_2}r$, where $2<p<q<r$ are primes and $N_1$ and $N_2$ are positive integers.