Adjacency matrices of zero-divisor graphs of integers modulo $n$

Abstract

We study adjacency matrices of zero-divisor graphs of ${\mathbb{Z}}_n$ for various $n$. We find their determinant and rank for all $n$, develop a method for finding nonzero eigenvalues, and use it to find all eigenvalues for the case $n=p^3$, where $p$ is a prime number. We also find upper and lower bounds for the largest eigenvalue for all $n$.