# Laplacian eigenvalues of the zero divisor graph of the ring $\mathbb{Z}_{n}$

## TL;DR

This paper investigates the Laplacian eigenvalues of the zero divisor graph of the ring _n, proving Laplacian integrality for certain cases and characterizing spectral properties and connectivity for various n.

## Contribution

It establishes Laplacian eigenvalue properties of zero divisor graphs of _n, including integrality and spectral radius, and characterizes when algebraic and vertex connectivity coincide.

## Key findings

- _{p^t} has Laplacian integral zero divisor graph for prime p and t2.
- Laplacian spectral radius and algebraic connectivity relate to eigenvalues of a vertex weighted Laplacian.
- Conditions identified when algebraic and vertex connectivity of _n graphs are equal.

## Abstract

We study the Laplacian eigenvalues of the zero divisor graph $\Gamma\left(\mathbb{Z}_{n}\right)$ of the ring $\mathbb{Z}_{n}$ and prove that $\Gamma\left(\mathbb{Z}_{p^t}\right)$ is Laplacian integral for every prime $p$ and positive integer $t\geq 2$. We also prove that the Laplacian spectral radius and the algebraic connectivity of $\Gamma\left(\mathbb{Z}_{n}\right)$ for most of the values of $n$ are, respectively, the largest and the second smallest eigenvalues of the vertex weighted Laplacian matrix of a graph which is defined on the set of proper divisors of $n$. The values of $n$ for which algebraic connectivity and vertex connectivity of $\Gamma\left(\mathbb{Z}_{n}\right)$ coincide are also characterized.