Eigenvalues and Wiener index of the Zero Divisor graph $\Gamma [{\mathbb{Z}}_n]$

Abstract

The Zero divisor Graph of a commutative ring $R$, denoted by $\Gamma [R]$, is a graph whose vertices are non-zero zero divisors of $R$ and two vertices are adjacent if their product is zero. In this paper, we consider the zero divisor graph $\Gamma [{\mathbb{Z}}_n]$ for $n=p^3$ and $n=p^{2}q$ with $p$ and $q$ primes. We discuss the adjacency matrix and eigenvalues of the zero divisor graph $\Gamma [{\mathbb{Z}}_n]$. We also calculate the energy of the graph $\Gamma [{\mathbb{Z}}_n]$.

11 pages